LECTURES IN ADVANCED CALCULUS

A THESIS

SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY

IN PARTIAL FuLFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

BY

NATHANIEL TORRENCE ~JHITE

DEPARTMENT OF MATHEMATICS

ATLANTA, GEORGIA

AUGUST 1960 .1

ACKNOWLEDG~NTS

The writer wishes to thank Dr. L. Cross, Head of the Atlanta University Mathematics Department, for

his inspiration and indispensable aid in making this

thesis possible. The writer is deeply appreciative to his wife for inspiration and encouragement.

ii INTRODUCTION

In the field of mathematics, students usually do not encounter or begin to develop precise analytical proofs which contain several members of the Greek alphabet until the student has en tered a course in Advanced Calculus. Since logical, accurate, analytical proofs are stressed here rather than mechanical manipu lation, the unlimited usage of the Greek alphabet and the vigorous treatment of proofs tend to present a problem of adjustment to the student. This thesis was written with the hope of clarifying some of those problems. Some of the more important theorems and concepts are presented with proofs that are sometimes abbreviated in several textbooks. The thesis is divided into three main topics dealing with differentiation, series, and integration respectively. Generally, definitions and theorems are given without examples since any stand ard textbook will contain a sufficient amount of examples to illustrate the theorem or definition in question. The work pre sented herein is usually studied during the second semester of graduate work in advanced Calculus. The writer hopes that the reader will find some of the more difficult concepts and theorems encountered In mathematics somewhat simplified.

iii TABLE OF CONTENTS Page

ACKNILEDONENT •...... e...... s.••.s•e••••••s•

INTRODUCTION...... -...... e... iii

CR~TER

I. DIFFERENTIATION...... •.....•••••••••••~• ...... 1

The Derivative •.....•.•••~••~•••••••e••a,•••a~,• 1 Rolle’s Theorem •...... •a..•.•••~a•••••••••,•••• 7 The Mean Value Theorem ~ 7 TheGeneralMeanValueTheorem...... 7 Meaning of The Mean Value Theorem ...... 8 Geonetrie Interpretation of The General Mean Value Theorem .. .-...... a . 9 Consequences of the Mean Value Theorem ...... 10 Ta~r1or’ s Theorem and The Extended Law of the Mean •...... •• 12 L’ Hospital’ s Rule and Indeterminate Forms ...... 13

II. SERIES ~ 16

Sequences •.a...... e..~a•..••••a•a•••a• 16 Series ...... a...... a...... e 16 o Partial Sum •...... 16 Convergent Series •...... •....~..•.•••• 16 u Monotone Se~zences •.,...... ,...... ~ 17 Bolsano — Weirstrass Theorem ...... 20 Series of Positive Terms ...... 2k Comparison Test For Convergence •...... 27 Comparison Test for Divergence ...... a . a.. ...• 27 The Root Test •.,...... 28 The Ratio Test •...... •...... •...... 29 e nu er 0 Alternating Series ...... • 31

Absolute Convergence •...... •.. .-...... 32 Domain of Convergence ...... ,. 33 Z Part.al Summation ...... 314. Addition of Ccnvergent Series •...... 36 1~ltiplication ot Convergent Series ...... 37 Rearrangements ~ 38

III. INTEGRATION •...... Li.].

Properties of The Riemann Integral ...... k2 z The Riemann — Stie].tjes Integral ...... k8 The Riemann - Stieltjes Integral as a Limit of a Sum ~

iv Properties of the Riemann — Stieltjee Integral .. 52 Scope of Riemann — Stie1t~es Integration ...... 5k The Fundamental Theorem of Calculus ...... 56 Bo~ed Variation ...... •• • ...... ••• 57 More Properties of The Rieinann Stieltjes ...... 62

~ ...... •••••••••••••••.....•••••••• 67

V ~NAPTER I

~iFEEREN~&T~P~

The De atthv~e..-A function y= .f(x) is said to have a derivative or be differentiable at a point X if and only if

~ 4~x) = f(x+Ax) 4 exists and is finite • The functional value -frx) defined by the limit is called its 1 derivative.

The above definition may be stated in another way, that is, the derivative of a function is the limit of the ratio of the

increment of the function to the increment of the independent variable when the independent variable varies and approaches 2 zero as a limit. Symbolically this means:

Ci), Given a functional value (*) ‘f ~ (X) and let X

be fixed • Let ~c take on the increment ~ Then y takes on the increment AY giving the

new functional value (* *) Yi-Ay = 4

(ii). Subtracting (*) and (* ~) we get ãf= -f (xi-6x) - -~(x)~

(iii.) Dividing by AX we then get ~X_ = f()4AX) f(x>, AX

1 J.M.H. Olmeted, Intermediate ~ New York, 1956, p. 85. 2 William A, Granvilie and others, F~lements of Calculus, Boston, 1946, p. 21. 1 2

Let t~X’O and the limit (if it exists) of the right—hand member as ~1x—. 0 denoted by j~, ~ 4~cx+djx)- fç~?_, which defines the derivative of + at the point Y~. note that

(1) If Ax and tly have like signs, then as the increment of the independent variable increases the increment

of the dependent variable increases and vice versa.

(2) If AX and AY have opposite signs, then as the increment of the independent variable increases, the increment

of the dependent variable decreases or vice versa. (3) Differentiability implies continuity but continuity

does not imply differentiability.1

Theorem I. Chain Rule.

If Y is a differentiable function of U and if U is a

differentiable function of X~ then ‘Y, as a function of x, is

differentiable and AY._ = . ~Ix du dX

Proof: Let be differentiable at u=1L0 f(x0~)

and let U~ f(x) be differentiable at ~ = Xo. As s. takes on

the increment ~ U wiil take on the increment t~ U • Hence

(*) ~ ~(u) and(* *) U f 0’) . Adding the increment ~x to e*) and ~u to (*~ *) we now get a new functional value

U) Y+Ay = ~(u÷ ~u) and (2) u+~u c (x+~x).

1 J. N. H. Olmsted, op. cit., p. 86. 3 ~ Subtracting C*~) from (1) and dividing by 4(( we get (.,) 4~I~

1—~-~-~ ~ - Subtracting (* ~ ) from (2) and dividing by a)( we get (4) ~ + .€ix) By combining (3) and (4) 4X we now get 4Y 4CL 6Y41LL ~Z . Therefore ~w~r 11X4L( C~X

~ j~j, and since is a continuous function ~X 4LL ~2X of X at x ~ then ar-+oimplies 4L1-. 0. Define 4U by

d~L ~ ~-c iiu~O 1~ 1

The-n ii~ e(4u.) = 1;in ~ ~ — dv _.~ aU-~’o cI~ ~

zc ~ ~ tIze~ froD, (~) we ~t A~ + ~ di~

4~ =ir. ~ne ÷ &(~LL)

Now 1/141 •4~ = urn dY 4~ ~. li~ 4CC gx -‘ o a x ~ ~ 4 X 4X-PÔ ~i~) 4~-)O ~z

Theteforc~ dv _ ciCC + ~. dci clLC dx Theorem II.

If y fCx) is strictly monotonic and differentiable in an interval and if f~~) a in this interval, then the inverse

function K is strictly monotonic and differentiable in the corresponding interval and = Proof: We proceed along the same lines as the proof for

Theorem I. Let (%~) y: 4Z~i and (~3 X cP ~). Then by

1 Ibid., p. 88. 4 letting (~) take on the increment ~y and (~ ir) take on the increment

,~( we get the new runetional value (1 v + a ~ = -~ (x ÷ 4 c) ~ and (2) X + ~ ~ = 4~ (y + o Y) Subtracting ~4t~from (1) and & *~) from (2) we n have (3)4y--f?(x-~~A.IO-.€Oc~) and (4)~4’c~y÷4V)-4~y)., If we divide (3) by the increment ~ and (4) by the increment 4 Y we get (5) ~f~.4r)—1’C) and (6) A.i.ar = ~(y+4y)- ~

Combining (5) and (6) we have ..~4L ~≤ ~4Y 4X ~ ~ — UK 4Y 4Y4) UY ~iy jx Since ~(Y)is a continuous function, then 4X-~ if and only if 4y~’o.

34~b)~C I’$ ax JIh~ _j_ = — I ~v~e ~! aY-~o ôy aX-v~ -~-~ Ii’.~ .≤~L.. d~1 ar ar-~ ax Left—sided and Right—sided Derivatives. Definition 1. The derivative from the right of a function 4 at a point~=a is the one—sided limit IIin 4f*h)-c~a~,.. 2 Ii J Definition II.

The derivative from the left of a function 1~’ at a point x

is the one—sided limit I issv S4~Z~’ h4) — 46t) L. ~‘ ) Definition III.

The right—hand derivative of a function f at the point K ~

1 Ibid. 2 Ibid., p. 89. 3 Ibid. I •iI~~

5 is the one—sided limit Ii~s~ Sf(a4 h~OL 11 )

Definition IV. The left—hand derivative of a. function at the point ~ is the one—sided limit I~ip~ f f(4ih) —4 ~r’)1, 2 ~1-~°

Definition V.

The right—hand limit of the derivative of a function -c at the point ~ is the one—sided limit -F c-~a •f Zr). ~

Definition VI.

The left—hand limit of the derivative of a function i• at the point ~ is the one—sided limit f1a). hi”t - .( ‘&‘ - ~—,. a

Note. A function has a derivative at a point if and only if it has equal derivatives from the right and frce~ the left at the point.

Theorem III.

If f6) is continuous on the closed interval £~, b) and differentiable in the open interval (a, b) and if -c (ic) assumes

1 Ibid. 2 Ibid.

3lbid.

4 Ibid. 6 either its maximum or minimum value for the closed interval [~a, &J at an interion point ~ of the interval, thea ~ o. Proof: Assume the hypothesis of the theorem and, for

definLteness, let f(~) be the maximum values of ft’c) for the

interval £~, 63 where ~‘ ~ ‘k. Consider the difference quotient

ay + a x) - c (~ for values of 4x so small numerically ax

that ~ .~. ox is also in the open interval (a, b). Since ~ (~)

is the maximum value of 4’(x), P~) .~ ~‘ (~, +4K) and ov~ ~.

Therefore ôY~0 £or4X~O and )~~O for JX >0. Dx Hence (in the jimit), the derivative from the left at

is non—negative and the derivative from the right at is non—positive. By hypothesis those one—sided derivatives are equal and must therefore both equal zero,1 Now, let .P (~) be the minimum value of 4~(K) for the closed

interval ~, b3 where ~Z ‘ ~ be-Consider the difference quotient

OX ~ ÷ 4*’)4X — f(~) for values of j~ sc> small numerically that~44Xis also in the open interval(cl,b),Since I~) is the minimum value of C(x)~ f(~) ~ f(~ .,-ix) andây ≥~ ~ • Therefore

~ ~, for ~r 4 ~ and LI ~ ô for 4 K > t’. Hence, (in the limit), the derivative from the left at is non—positive and the derivative from the right at is non—negative. By hypothesis these one—sided derivatives are equal and must therefore both equal zero.

Thid., p. 93. 7

Theorem IV. Rolls’s Theorem.

If a function f is continuous on the closed interval ift, b) and if f ‘~) exist in the open interval (~j b) and if -e ~ Cc~ &) C,then there exist a point ~ ~ (a~ i) such that

Proof: If ffr)~o on [~,1,Jthen the proof is trivial.

If 4”(x) ≠ C then somewhere in the interval L~z bJ it is positive or negative and reaches a maximum or minimum there. Therefore

Theorem III applies and the proof is complete.

Theorem V. The Mean Value Theorem #1. If ~(~) is continuous on the closed interval 1.~a., bjand

exists in the open interval (a, b) then there exist a point ~ ~ (a., b) such that c~ i~) ‘(~t)t CC~) (i,~- a) or

~ ______~ assuming ~ ~4 1 and 2 ‘b. This is actually a generalization of Role’s Theorem.

Theorem VI. The General Mean Value Theorem. If 4ft) and are continuous on the closed interval

~ i3 and if f’c~r) and g ‘~k) exists in the open interval (~) b)~ then there exists a point ~ ~ 1~)sucb that

rc~)[~)- +(a~] - )3.

Proof: Consider the function F& defined by FC~) f~) - where 1< is to be determined. F is continuous since fOQ and ~ (‘9 are continuous on the open interval (a, b) and F2x) exists in the open interval (~, b) We want I~ such that f(a) — K ~ ~a) P(&) -

That is — f(b)- c(a) S

Suppose that ~ (1,) ≠ ~ Then Ic ~~ Thus in this case, the hypothesis of Ho].le’s Theorem, relative to~ F are satisfied. Hence, there exists a point ~ in the open in terval (4 6) such that F ‘(~) 0. That is, f “(‘4 )- w g -~ ~

Hence the result. If ~ = ~ (b) then we apply Roile’s Theorem to the function Hence there exists a point ~ in the open in— terval~’4, b) such that ~ t~. and the theorem again holds. Q.E. D.

Meaning of the Mean Value Theorem.

— —I

a 6 Figure 1.

In figure 1, let slope .e~ ~~ and slope £ - f’ is tangent to ~ and parallel to I and £ - parallel to 1 implies Between the derivative and the difference quotient there exists a simple relation which

is important for many purposes. This relation is known as the Mean Value Theorem and is obtained in the following way, We consider 9 the difference quotient -~4X — oi a function f(r) and asewne that the derivative exists everywhere in the interval ~ K ~ 6, so that the graph of the curve has a tangent everywhere.

The difference quotient will be represented by the direction of the secant. Let us imagine this secant shifted parallel to itself.

At least once, it will reach a position in which it is tangent to the curve at a point between a and b~namely at the point of the curve which is at the greatest distance from the secant.

Hence there will be an intermediate value F such that f?~) : ______This is known as the mean value theorem of Integral Calculus.

Geometric Interpretation of the General Mean Value Theorem.

~,U.), f(~)}

figure 2

In figure 2, consider the curve C in parametric form:

1 R. Courant, Differential and Integrgl Calculus, New York, 1942, p. 72. ~; : ~ where a ~ & • ~ ~‘°p° and slope 1’ = ____ . i’ parene to 1 implies dx t~ ~ç)

J~52 = (b)-c(a0

Let 4 be continuous in a neighborhood L~ of a point X: ~ and let f “(ic) exist in that neighborhood. Then there exists a. point ~ (~ b) jwhere 1’ £ U and a ‘b such that f’(’%) b)—4~!~1).,, or .ç~4’(a) +f’(ç)(b-i). 5~t hb-a. Thefl b~ 114a and

e h. ~e€r~c ~‘ ~ I. Henc~ f(b) ~t~) +f’(’~)(b~i) becomes ~(i÷ h) = -~) + h (F’~ + e h) ~, he rc. ~ ~ I,

Consequences of the Mean Value Theorem.

Theorem VI.

If a function has zero derivatives throughout an interval, then the functicn is constant in that interval.

Proof: Let the function be denoted by-c . Let -P “tic) throughout the closed interval fa, &J. Suppose there exist:;two points

C. and d belonging to the closed interval £~, i~) and ~‘ ci such that ~ .ç (tO. Then by the Mean Value Theorem there exists a

point ~ such that ~ ~-~: ~ ~a. But this contradicts

the fact that P ?4d) 0 ei~ Ca, 62 Hence f c~) 4~ (4). Q.E.D.

Theorem VII.

If fL”) and ~(x) have the same derivatives in the open interval then -ft4 and differ at most by- an additive constant in (a~&). 11

Proof: Let • (‘,c) ~ ~ (~, b), Set k 6c) -f Cx) - ~I, O()

Then h’ (~C) ~‘ ‘(r) — ~ n (a, b). Hence by the previous

theorem, h ~ ~. (where L is a constant).

Theorem VIII. If f?x) exist in ~ and if f’(x) ?b -f beti ~6c) ~s

monotonicafly increasing (decreasing) in (a) b)~ If -~ ~*) , (‘ c), then -f(x) is strictly monotonicafly increasing (decreasing) in (a, b).

Proof: Let X. and ‘c~ be any two points in (a, b) with

~ ~ By the mean value Theorem there exists a point ~ belonging to (Y~~ X~) such that 4~(x2) (fr1) -

He.n~.e 4?(ç) = -f(~) - 4~r1) — K. and the conclusion can be read, from this expression.

Theormm IX.

~f 4? has non—zero derivatives throughout an interval, then 4~r) has the same sign throughout that interval, and hence

is strictly monotonic there.

Proof: By virtue of Theorem III, it is sufficient to show that -c ‘(i) has the same sign throughout the interval. Suppose

f’(x)~D in (4)b) where d~ b Suppose there exist points (~,d)~ where t~ 4, which belong to the interval (“i &) such that 4’ Cc) and -c(d) have different signs. By a previous theorem -P(x) has a maximum and minimum on Ct,t1J Let be a point where ~&) takes its maximum and a point where fO() takes its minimum and ~ ~ 1(. since PCy) ~ and do not belong to the interval

(~) 4), for by a previous theorem 1~ ~‘~) b which cannot be. Hence, either and ~ ~r c~ = 1i~_ ~2 ~i Case 1. C. is a maximum. Then c’(c.)~o ~nd f’&!)~O.

Case 2. ~ is a minimum Then ~‘c~cI)~ ~‘nd 4”C~:)~t)•

In both cases we have a contradiction. Q.E.D.

Ta~i.or’s Theorem and The Extended Law of the Mean. Definition. A function 1’ is said to belong to class ~ ~ (a, 5)

if and only if 4”(~) exists and is continuous in (4~ b), and vi~ 0

Theorem 1.

If fb is defined on a closed interval and if

exists in (~)b) and if continuous on Ea,bJ,(and exists

on . b)),tiien 4’ (b) ~ ~c~z + ~E ~ (b -~)~ + R, W~ ~ R ~~?(b-a3~’ r—~ ic. (ii+i)I

and ~

If ri: 0, t hen f(&) r ~ ~ (b-a~ • ~ re4uce~ to

the Mean Value Theorem, for then we have~ ~ (~) ~ + ~ ~ o+i)

So feb) : 4’C~’) -~ 4”C~)(b-c?) and this is a generalization of the

Mean Value Theorem. Proof: Let I’1 be defined by

% + ~ f~’~) ~b-4~ + ______~I 13

Set 4’(~ — + c(~) + ~i + ______KI a ~ t ~ 6. We wish to show that fri c~’~ r ~ 6~)’

Observe that f(a) f(fr) ~ Moreover F(~) exists in (a) k). Applying Rolle’ s Theorem, we see that there exists a point

e (ci, ~) such that 1~

~ut F’(t) c’(~) -t- ______K’, — f~~z). (b~t)’~j

~ J~4 (t~-tY’ = ~ (6.~)ra — jkl (b-t)11 -

Thus (b ~‘ (f~‘flt) N) F ~t c rc t ~ (~

Thus F’(~)~c implies F~’~’~)fri = and this means

M. Q.E.D.

L’Hospita]. ‘a Rule and Indeterminate Forms. Any expression of the form, ~ ) ,

00, 0 00~ etc., is called an indeterminate form.

Theorem XI, L’Hospital’s.

1ff and ~ are differentiable in fL 61, ~f ~ ~x) ~- 0

(4~ 1,) and it

(i) lips f&) 0 ~ ,i~I ~

(uj)~ (i’s% -~(x)~oo arid hpi ~(y)o~

(iii). (jglt fZ~~ L x—4 b g’Lx) — ___ — L f~?) - L ~-1-,whcr’c ~

14 thenli~i f~) = L ~-÷r ~j Suppose ~‘(x~ ~ ~ in (a~ L’) and is continuous on r4~ bj. Then we show that ~ (~*) ~ ~ Proof: By the Mean Value Theorem, there exists a point t (~,t~) such that 1(b) ~ ~

That is’~Cb) ~4) l3~t (t,-.4) ~t

(since ~ < , ~nd gc~ O~ ~(cv~.e gcb)- ~ So

Theorem XII.

f and ~ belong to ~ j~, i~J, ~ ~r) ~ t~’ In (~Q) b) in, pl,~cs that there exists a point ~ c(a)b~sueb that ~ = ______

Proof of L’Hospital’s Rule:

1. Under hypothesis (i). We assume b to be finIte and

Sinc~ urn P(r) to ~iid /~ o LV~ t*t~ ~set ~(b~o

and ~ (b)~a. Then ~ and ~are continuous on Applying the General Mean Value Theorem we see that

there exists an ~ in (“~) b)~ where x ~ (~ i,) a #4 a b

5~e~1, fbA* -f(&)’ f6) ~ £ #11) t~) gz-x) Since Iii, Ls there exists a ~4 ~ such that ~ -i 6 ~ ‘c~x)

g~x)

Iie.tee ~ f (r)~~~ ~ L ~—~b tO’•) ______fe~x)— fr°-__~ 0

15

2. Under Hypothesis (ii). First given any E >

choose ~ such that ____ — L ~‘ ~ 4~Or 1~-.’’ ~ ‘ g Set x. b - • Then by the General Mean Value Theorem

there exists a point R in (~•, )t) such that

f (x) - 4~ = ______- — ~ I )-g~~) I gcx) I

Then observe that — L fCx)-~c -L ~ —~6~,) Nof~ = ~ h(~) ~ j ~ ~ t~ b-~ ‘~ ~ b, *h~ ~c~k)4l g cx) as X—P~by virtue of hypothesis (ii). Hence there exists

an ~, in (YD~ 1.,) such that b~K) ~ j ~hc( (~ttii i( ~ forai.1 ~~X1 .€K~b )ff(x”\h~ h(x).LI4 lc(x) Then i~-_----- — ~%g Cx)

+ ≤

Wt1~h4VCr r,~ ‘ x’&. ~ - ~ 1:fJ~? — LI h(x I~(4 t ~.L ~f±f±!~~ -L’ .c_[tI1~I € < I+lLI]~tkewcv€?. -‘ I ~ ~ b. ~ — Jc—~b 2(* Note. If b~+.e then we may define the functions F and~as follows: F~Q P(.4) ~ 6~(-~). ~ as

hience Ilr1~. __ - 11$,. ____ - X-’p3~ ~L) X~O4 ~(jc)

This case is reduced to the case just discussed (i) where 6 is finite. CHAPTER II

SERIES

Sequence.

Definition. If a13a2,.~. an,... is a succession of terms formed by some fixed law then we call it a sequence. Briefly, a sequence is a function defined on the positive integens. We denote the set (at, ~.i, -. by an and cail it a sequence,

where ~ ~ ~ ~ ~ ,~,... ~ . Each element of the sequence is eafled a term of the sequence, with 4 n being called the general term.

Series. Definition.

A series is the sum of the terms of a sequence. That is,

ir (an~ is a series then (a1 -i-~+ ~4.—-) is a series which is denoted by ~ If there is no possibility of confusion we writer an for infinite series or, more briefly,a series.

Partial Sum.

Definition. The sum of the first tI terms of a series denoted by,

S~ z ~ a1 ~ - - 4~ is a partial sum.

Convergent Series. Definition. 16 17 Every finite series converges and every infinite series con verges if and only if ~im ~ exists and is finite; otherwise the

series diverges. Examples.

1. Consider the series VI~ 1, ~

S,~= l+2t3+-- 1-n !Ic~÷I) , an~ ‘trfl Sn ::-foo. 2 In this case, although the JIrI~ Sn exists, the series h 4 PG diverges to •

2. Consider the series ~ (‘.-i)~’ ~ I— I + I I + I ~ + ~o”’J. Isr~i 3,, does not exist in this case and hence the series does not converge.

Theorem I. The altenation of a finite number of the terms of a series does not affect convergence. The sum of the series, however, is affected.

Proof: Consider the series ~ and assume convergence. Suppose rn terms of the series are altered. Since ~ is finite there exists a 4~1 fl?~ I such that the terms 4n~forA’$ are not changed. Thus Z ~ —, ~ 2~.

is finite and hence it converges. Let S11 a, ~ +4~ - arid ~ ~ ~ -~- ‘~€~ * - -. ~ ~ Sn = Sd-, ~rz4

iini (Se-. ~ ~ ig~, S., —5g_, - Hei,c~ ~ S,., eg~1~ and i~ 4’.,A+t, Q.EP

Monotone Sequences. Definition. A sequence LanJ is said to be monotone increasing if for 18 each h we have an a ~ ~ a~~1,where ii I, a.~ - - The sequence is said to be monotone decreasing if a~ ~ Un,., W~ter€~ P~I,2,S,.. If the sequence is monotone increasing or monotone decreasing it is cailed mónotonic.

Definition. A sequence is bounded if and only if all of its terms are contained in some interval. Equivalently, the sequence {a113 is bounded if and only if there exists a positive number P such that 18i,l~p for aR h. Definition.

Let be a sequence. Then the number 1. is called the

].imit of the sequence, written t.~i, ~ L~ if and only if for any

~ >D~~there exists an N such that jar~ — L I Cor all ri > N.

Theorem II. The C~u.cL~y C.u*e*ion foR. ca~ve~,~1~e of ~ %e~ewce. A necessary and sufficient condition for convergence of the

sequencef~~J~s,that, for any é~ ~°there exists ai ii ~Ke) such that ~ ~ for ~~(( ri > N and w. > N, Proof: Assume that the sequence {Un~oonverges. Let L be the limit of the sequerice{à.~3. Then there exists for any~ ~ an N Nt~)

such that ~ - L I 4- a .~4 j a~, - ~ € ~ U ,,~ ~,, > V. Observe that li2n~?,n1~ (a,, ~ for all i~r~’ ?N. Hence the condition is necessary. Assume the condition that (c7n-dn,I~~ for ~fl rIjrr~ >N.

Given é~> there exists an Nt W(~) such that un- a,,~ ~ for all

~ ~ ~ aU ri> 19 Hence the sequence {q ~} is bounded. Therefore, by Theorem 114, the sequence~q~ has a limit point. Call this limit L. Suppose there

exists another limit point .1. such that 1.~ ‘4 /~ t) • Then set /~t ‘-b/ 3 ~• We laiow that there exists an N~,W~) such that /~y~/.~f nwi/am -4”4~

for all ~ ~ > N(4rhen observe 44~L 3~ /h~-4/-/i’.~ i~ ~ t41,~y -~ /

for al1J7~m ,,i/. Tbu~ we have that 3E ~ which is a contradiction. Hence the condition is sufficient. Q.E.D.

Theorem ha.

A necessary condition for convergence of a series is that the limit of the ttherm tends to zero.

Proof: Assume that t1~e series~Ø4,converges. Then urn a~ ~ • Consider the partial sums 3~ ~q Observe that I~bO

— . Suppose that L is the limit of the partial sums. Then him G~ i’m ~ —S~.,J ~ 3,, 1,m 6~.., :.4_ b O. n-. ~O T4,-ç~p~ ‘im d, ~ Q.E.D. The above condition however is not sufficient. For example, consider the Harmonic Series ~ * Note that ~

(i) The series ~ .1 _/L,~/ -

Hence the series ~ 1,,(.) , -

The series ~ - which diverges to

Therefore, since the series j -~ is greater than the series 64 represented in (ii), then the series ~ diverges. 20

Theorem II b. Every monotone increasing sequence, bounded above has a

limit. If a monotone increasing sequence is not bounded above, then it approachesf OQ as a limit.

Proof: is not bounded above.

Since is not bounded above, there exists for any positive

number M.~ however large, a number ~1L such that ~r1~ M. But since

is non-decreasing (~ ‘ M for all ri> fl, , ~ ~ = -~ 00. {Q1~ is Bounded above. since{Q ~ is bounded above it has a least upper bound say 1~

Then given ~ >o and arbitrary there exists an such that B - €

< ~ ~ ~3-~ 4a~+ince’~L~ is non-decreasing we have B ~ ~ ~ ~c for all fl> fl6 ~adding -Bto the

inequalitywe get -~ Vh oR\(~-S\<~

~ ~ ~ Theorem II c. Any bounded monotonic sequence converges. If the sequence

monotone increasing and if (~~~for all V~ +h~i~q~converges;

moreover, if q~, B then ~ ~ 6 ~. I’ -~. Q~L v~. Proof: Since the sequence(~\ is bounded above then, by Theorem IIj,, it has a least upper bound B and for~>ô there exists

~ such that ~ ~ ~c E for all I~> ~ ~ We wish to show that Q ~ ~ B’ Observe that given E)Othere exists a position integeii~ tJ

such that (~ S-~.Therefore E-€ c~c~ ~, q~ ~ ~. t3 ÷ ~ ~•

ô~. k~-~1’~E ~. qj~ ~14’

Theorem III. Boizano — Weierstrass Theorem. 21

Every infinto bounded set has at least one limit point. Proof: Let S be an infinite bounded set. Since it is boi4ded, there exists a closed intervalZ such that~r. Divide ‘into two equal parts. At least one of these parts contains an infinite number of elements ofS. Let this part be denoted by.1. Divide into two equal parts. In one of the two parts there must be an infinite number of elements ofS . Denote this part, .t~D

Continuing in this way, we obtain a sequence of non—increasing closed intervals E~ ~ i Each of the intervals contains infinitely many elements of ~ and the length of the interval tends to zero as ,~ . Therefore (~ ~ in is a single point. Denote q this point byh1. Now, consider a neighborhood ofq. ie. (~i-~ )V~+~.)w There exists an interval I such that X,~ ~ The interval :t ~

contains infinitely many elements of ~ and therefore (‘~ - ‘_) n * contains infinitely many elements of S . Hence ~ is a limit point of5. Q.E.D,

Examples.

1. Consider the series ~- ~\ )i £+ ~* -

In this case, although the ),,,.‘ S., exists, the series

n -~ diverges to ~- ~.

2. Consider the series ~ - j

case and hence the series does not converge.

Theorem IV. ~22

If a series converges then its sum is unique.

Proof: Suppose S~W. and where L 1. . Let E IL — I..’ I. Then ~ i—~ta\-i ~Si~-t.~l•

By assumption there exists Q1~ ~1 i~) such that ~ s&’ I ~

- for all ~‘> t~. Therefore 1, + ~,. - -~ ~ ~1) e~>N, This is a contradiction. Q.E.D.

Theorem V. If all terms of a sequence, from some point on, are equal to a constant, the sequence cotverges to this constant.

Proof: Any neighborhood of the constant contains the con stant and therefore all but a finite number of the terms of the 1 sequence.

Theorem VI. The sum of two convergent sequences is a convergent sequence, and the limit of the sum is the sum of the limits:

‘~ r~i~-~ ~j;sib,~.

This rule extends to the sum of any finite number of sequences. Proof: Assume and ~ ,Let ~pO be given. Choose i~ so large that the foilowing two inequalities hold simultaneously

for h ‘14. 1 ~ . 4f~ ~. Then for we have

i~1 t~(q~~~) ‘(~~-b)1 ~ ~ j~

~

1 Ibid., p. 34. 23 Theorem VII. The product of two convergent sequences is a convergent sequence and the unit of the product is the product of the limits:

~ ~b~) j,i.~ ~p3. j~fl~jr 3 +~O This rule extends to the product of any finite number of sequences. Proof: Assume a~-~q and ~ wish to show that Qe~,,.

or equivalently that ~ ~ -~ ~ By addition and subtraction of the quantity ~ 1~ and by appeal to Theorem VI, we can use the

relation ~ b ~ — ~ (Q~-C1)bh t q (hi, -b) to reduce the problem to that of showing that both sequences

- q) ~ and ~q(h,. -~) converge to zero • The fact that they do is a consequence of the following lemma. I~E-Au:f1~ converges to zero and’ ~. converges, then ~4i. ~ converges to zero. Proof of lemma: Since any convergent sequence is bounded, (àa3 P the sequence ,~ is bounded and there exists a positive numberA such that ~ ~? for all~). Given ~D choose N so large that

\‘~. ~ for all ~p N. Then for ~ > ~ \ C!. ~ ,.~ ~ ~ \ ..\~, ~ c This inequality completes the proof of the lemma and hence the

theorem.1

Proof that any convergent Sequence is bounded: Assume ~ -~ q

and choose a definite neighborhood ofq, say the open interval

Since this neighborhood contains ail but a finite number of terms of a suitable enlargement will contain these

I Ibid., p. 36. 24 1 missing terms as weil.

Theorem VIII.

The multiplication of the terms of a series by a constant ,

does not affect convergence or divergence.

~roof: This is a consequence of Theorems V and VII.

Theor ~m IX. The Geometric Series.

.~ series of the form ~ ~.

~ where ~ is cafled a Geometric Series. The Geometric serie~i converges if\y~~ and diverges if ~ Proof: Recall that

~ I~Th~i, j.i,~SN~_— ~3~r~’s1 ‘ ‘~ ~%%~ ) \,ç~

If ~ ~, then we have two cases. Consider the case where

This c~ase is clear so we consider the case where~Y~j. If

~Y1i tk;en ~ ~ Hence the Geometric Serje~ diverges if’frj>1

Series of Positive Terms.

Cpnsider series of positive terms or more generally series of

non—negative terms, then the following theorems hold true.

Theores I. A series of non—negative terms is convergent if and only if the pa ttiaJ. sums are bounded.

1 p. 35. 25

Proof: (i) Assume the series ~ 4~ Then for ail ~ .~ Therefore the partial sums are bounded. (ii) Note that our requirement that (h~o implies

that ~ ~ * q ~ s~ Thus we have a bounded monotone increasing sequence of positive numbers and by Theorem lib, every such series converges. Therefore the series is convergent. Q.E.D.

Theorem XI. If ~ a then the series~ ~ converges

if and only if the series ~ ~“ converges.

Proof: Set ~S)~ ~ ) (1) Consider the case where Then ~

(ii) Assume that fl Then

~ ~. ~ - ..4 ~ IIcvce ?~ The theorem follows now by applying Theorem’X and considering the above inequalities.

Theorem XII. The series ~ converges if and only if there exists an ?~ such that \ ~ ~ ~, ~ v.~., >

Proof: ~ q~ •~ .4O,~. A~viJ Sm ~O~-tO~t Then by the Gauchy Criterion, given ~ ~ there exists an

Id ~ ~) such that /~ - ~ ~ a I Observe that O~ h,aw>’V. 26

~ ~Ncc S0i~.c,~ ~ Theorem XIII.

The series ~p converges if 9>~ and diverges if Proof: If then the series diverges since h ~ —b #

If ~ ~ we apply Theorem XI. Thus the series b~ a ao~ k ~ (~~‘ .~I.. 2~ ~ which converges if / and

if and only if I- f’ ~ ~ inplies P’/. Thus the series ~

converges, by Theorem XI, if ~ But the series ~

diverges if (\-~)~-~. ~ Hence, by Theorem XI, the series ~ — diverges if ~ Q.E.D. n~

Theorem XIV. The series ~ ~ converges if P>( and diverges

if ~

Proof: Consider the ease where P≥~ . If P>o Theorem XI

applies and hence ~ 2 ‘C

~0 ~ ~ which converges. If P).b then by applying Theorem II to the series we see that the (~Q-M:z.)P .~çP series diverges. 27

Definition.

The statement that a series ~. ~ dominates a series means that~Vi\ ~ b.. for every positige integer n.

Note — ~ny dominating series consists automatically of non

negative terms although a dominated series may have negative terms.

Theorem XV. Comparison test for convergence.

‘. C where 1~• i~ (a positive integer), and if the series ~ c,~ converges, then the series converges.

Proof: Since the series ~ c~ converges then given

~ >ç~ there exists an ~ ~ t~) such that for Y~ ~ ~

~ Ii ~C ~ 1c~ ~ :~:. ‘~\ ~. ~ - €

Whenever _~ >~j~ ~v, z-o • Therefore converges.

Theorem XVI. Comparison test for divergence. If for bi~.J~( and if ~ ~ diverges then

diverges.

Proof: This is a consequence of Theorem XV. i~ .e. if

converges, then ~ ~ diverges. This is a contradiction. Examples:

1. Consider the series _S___—__—_

Observe that for every hr~ • Hence, by

1 Ibid., p. 212. 28

Theorem XV, the series ~ ~.ii~— converges

since the series 5~ ~ converges.

2. Consider the series ~ — — —

Observe that ____ ~fl~

Theorem XVI, the series .1... diverges since

the series ~ -S.-. — ~ i.. diverges.

Theorem XVII. The Root Test.

Consider the series % ~

If (a) ~ ~ co~vergea;

(b) (~~j ~ diverges;

(o) ~, =. \ no information is given.

Proof: Suppose 6~q . Let be such that ~ ‘4’~ I ~ Then there exists an index such that ~ ~ ç~ all~’~, Hence for all Therefore converges since ~ /3~~ converges.

Suppose ~>I . Let~’be such that ~t4~ ~. Then there exists a subsequence~Q4~3 of the sequence [Q~~f and a A~ç)such that

> .~ -j~dM~ Q/f ~?k ~/j”~ #~‘ivtc /O~j >~

If ~ it is sufficient to consider the series b~ 29

and ~ • But observe that ~ ~Ji~ ~ 14, k~ ~ Therefore no conclusion can be drawn if

Theorem XVIII. The Ratio Test.

Consider the series ç~4 Q~ 1%~ ~ ~ If (a) P~j the series converges;

(b) -,~ for v~ > *~then the series diverges;

(c) (~; ~ — ~ then no conolusion ~ -.~ ~ ‘e~-~’~~ —1 — ~I% ~ can be drawn.

Proof: (a) Letfl be such that ~ Then for

there exists ~ 4t,~%(6) such that ______fan~i/ h >iV. /LIAf

~ 1c41,lI/ £ ,S ~S ~ ~ ~n/ Hence /a~ ~W~/4/ Therefore, by Theorem XV the series converges.

(b) ~ ~ ~pliLS ~

7W~ S~%/~ /3 Div6rp~~w~. Cc) Proof of (c) follows again by considering the series

~ ~ Observe that ~ 41 But the series diverges 4~

and the series ~ -~ converges. Hence no

conclusion can be drawn. 30

~ample a:

1. Consider the series ~... —

Observe that ~ Therefore the series

~ converges, by the Root test.

2. Consider the series ~j \‘\~

Observe that ~ —~ ~. ~.

Hence this series converges by the Ratio test.

The Number

Definition.

Let ~

Theorem XIX.

Proof: Set S)’%) ~4~4 3% - ÷ I Set4~-4-~. Observethat1~~(~

- ~ ~\-~ ~‘ ~ •k ~ (‘ - - - -

~ ~ e. ~ ~

~f N ~ ~ ~ ~

Hc4ixed and let p~-~øa1Then ~ ~ ~~se that

- ~)1 t I ~

- ~e4i~6~ ta,,, ~ 1 j 31 Theorem XX.

~ is irrational. Proof: Supposee.~. is rational. That is, where

and are intege~s greater than zero. Consider -~

z ___ -

~ ~‘ -‘. -k -

- ~4~’) ~ ~ -t •~::~~ ç’~.4\ — •~S_ i~%\.. — ~ -~. Hence we have ~ . Now, making use of thi~ in—

equality-, observe that c - ~ That is ~ -

But %~ e. q~! ~- ~ p is an integer, and

is also an integer. Hence %I~.~e s,.t, is an integeV~(~where O(~(,which is impossible. Q.E.D.

Alternating Series. Definition.

Suppose an ~ where ~i ~ ~ Then the series

~d ~ ~O~1.q~q3 ~~4_Q~Q~• is called an alternating series.

Theorem XXI - The series ~~ ~ ~ converges if;

(i) Q,~11 ‘q~.

(ii) ~ ~ b.

Proof: Consider (q~~ ~ t (q5 a~ .~.

-* (Q~,4~ -Q~).

Observe that since ~1> y ,,, ~ > Hence Zfr7l ~ ~ ~ .e~ c~s-/~s. $~E~A~ -(q~-Q))_(q9 -‘~.)_ - - 32

~uQz~1~. S~,’icc L%iv..3 ~L, ~ ~. This means that the partial sums are bounded. In particular, the I,,,, 5,,,., ~ Q,. I, -,

Note. S~1, S~ + ~ ~ I’m Oq >~- .5~

and hence o ~. By Theorem lib we have A; ‘.‘i Cs,.,~,,., - I1-7b0

~ 4,cr;shi~, S~1,t~ ~I~”

- Therefore ~ converges. We

set ~ ~ and claim that S - s ~ ~ Note that c~ ~SN-~ c~ ~ Howev~r, from abcve we have that the sum ~ ~ .4 C%~4~~• ~ \~-Sb~\.L G~4’~j £~~U ~4

Absolute Convergence The series ~ Oi~ converges absolutely if and only if the

series ~ ~ co~verges. If the series ~ O~ converges, but does not converge absolutely, then the series ~

is said to converge conditionally.

Theorem flu. Absolute ~onvergence Implies convergence.

Proof: Assume the series~ø,~converges.~ Note that a series

Q~ converges if and only if for ~>o there exists N~ W(Ej

such that ~ ~ for all Y~r$~> t~ ~ Therefore for 4’> (~

there exists ~ g4 (~ such that ~>

X~e~cE \~O~ic\c~ ~\q~\ S~r~ ~ >~vy~> ~, 33

Domain of Co4vergence.

Definition. Let be a sequence of complex numbers. Let ~ be a

complex number. Then the series C,~ ~‘~‘ is a power series

and converges on diverges accordingly as ~ belongs or does not

belong to the domain of convergence.

Definition.

Domain of convergence implies the set of all ~ for which

the series ~ ~ converges. The domain of convergence of a

power series is a circle in the interior of which the series con verges. The series diverges in the exterior of the circle. It is not easy to characterize what happens on the boundary of the

circle; therefore, to include all cases, we consider the whole

finite plane as a circle of finite radius and a point as a circle

of radius zero.

Theorem XXIII.

Set ~ Put ~ * For ~= L~

For ~ &o.If /~J~g,the series converges. If

the series diverges.

Proof: Put q ~ ~N By Theorem XiTII we have ~ ~ ~ Hence the results. Q.E.D. 34

Examples:

1. Consider the series ~ ‘ Observe that

and n — _____ — ~ .~~AE?4 OV~•%%: ~ . • O~Ib Thus we have that ~

• N \ - •~ e~ ~ ~ -~ ~ = -~ ~• Hence the series converges for ail j~[~ ~o

2. Consider the series~y~’~~’ Observe that j~ ~Jj~ ~

therefore the series converges for ~ 0.

3. Consider the series ~ ~ ~, Observe that ~ , therefore the series converges for aU~~~4 and for no points on

the boundar~.

4. Consider the series ~ . Observe that ~ ~ ~~JI _~4_~ ~

Thus the domain of convergence is \~. \ .~

The series does not converge at ~i but con

verges on all other points of the boundary.

5. Consider the series • Observe that ~ therefore

the circle of convergence is ~ • But hence the series

converges at all points of the boundary since \.~ \ ~ ‘,&.\= vi’ ~‘ ~

Partial Summation.

Tl~orem XXIV.

Let ~ and ~, ~ be sequences of real numbers • Set

~ %ç ~ ~ ~ (~. 35

A b~) ‘.s 5uPz~ Z 4nbri — ~ - +

~ i4p.i bp.

Proof: Observe that ~ ~ ~ — ,q~.-,) 1, ~ l.~p

Z A~’~ — and, ~-

~. ~ 4~b~ ~ A ~

where “~ fl-a and vi,,’÷i. Since these indices are dummy we may write that the above series ~: A~b,7 ~ ~A,,b,1 Z 10 p-i

= ~A,,b,, -,9~b~ - + ~ ~÷,J

z ~ ÷ d4c~ ~ — ~ Q.E.D.

Theorem XXV. If (a) 4~ of forms a bounded sequence. (b)b0~ b1 ≥ k~

(c)’~ ~,, ~c, Then the series ~ converges,

Proof: Let (4>0 be such that j .4 ~ ~ fri for all rz. Let N (~) be given arbitrarily such that br~ Then observe

that for ~ > ~ we have 1 ~ ~ b~.,

~ — b~ ~ -~- A~ b,~ - ~ b,. j ~ M —

-~ 4 ~bpj ~i2ML).p ~ ~ ~‘n ~

Theorem XXVI.

If (a) L~., ~ ~ - - - -. 36

I Iuw ,‘ -. (b) LII —

(c) Radiizs of convergence of the series ~.. Ct~ ~ is j

Then the series ~ ~ converges for all i ~‘ii I ~ 1 except possibly at the point (~l 1.

Proof: Set Are ~‘“ and &r, c-n. Then

, fi~ 4f1 ~, = ~aK1 1~~ei ~ I-1 therefore for all £ on [~1iexcept ~ Hence (i) a0÷a, 4a3 bounded for all ri. ÷ - - - - - . a11 is Ian’, I2r~ (ii) b0 ≥ ~ b2 ~- - - (iii) •b7 —~ ~ Therefore, by Theorem XXV, ~ converges for all

3on I~(~i except possibly ~ Addition of convergent series.

Theorem XXVII. - If the series a~., and the series ~ are con— I,~O

vergent series and if ~. and ‘4 are any constants, then the series

:j~ ~ converges. Proof: Set ~ ~ 2~ +~bk.Then the series

~L’~ ÷~j~) ~ <(t(~utsiiice

Li~a~, ~k ared 111)1 ~ exist and are finite

we have that ii g~ ~-I( exists and is finite, ie. ,1--~4. Ic- ~

~0(~~ih~Ifr ~ converges. If we set A,

and kr~~ ~ ~ we have that ~LA +.~16. h —~ ~o b 37

L.t1, ~ ~ + ~ ~A ~ ~ a Multiplication of convergent series.

Definition. Consider the series ~:a~ ~ b,, ~ •10 I7~b 00

Then the product of ~ and ~. bi, is the series ~

The product of C,1 is defined because if we multiply the two series together we have (2 ~ii ~ ~‘) (~ b,1 ~

~ +(a.~, 4~.b~)e -i- (~b~ *~b.

~ ~o ~—I But the convergence of two series is not sufficient for the convergence of their product, For example, consider the series .0 * Z (-‘~“ -i- - b~ s~ 1- + I — + - - 00 ~ \~u ~~+‘ SotI,at ~ E~i.tt -K4-I)(I(4-i)

; (~i. + — (~i - ~ ktue j~11Lt c-~ ,~(k*I) .~ ~‘.-i ~1~0 the 5lLa term of the series~ does not tend to Zero. Therefore the series %- diverges.

Theorem XXVIII. If the series (a) ~ a1, is absolutely convergent, HtO .0

(b) t ~. ~i ~o 38

(c) ~ ri.

then C~. s 13, iv ~ -e (I~ ~* -

Proof: Set l~,, b~- ~,, + ~*---+ ~ S€t ~6 13- 43~.

Then observe that C~ 0o 1~o +(QDbs +4’, * - - ~ (9ô 6n +% ~ I- - -

+ ~ &~~) (a I3~ ~- a, £3~, ,- - - - ~ 5,’) - ~ -, a,

Note that Iii~s. ‘4k 13 A 13. Therefore it is sufficient to show ~ -~ ~ ~ Q~ I1-> ~~O• Since the series ‘5 4~ is absolutely convergent, ~ I4~)

Therefore, given ~ O there exists an IJ ~t(≤) such

that i~ L~ é for all ~ > N. Hence, for all v~? N we have

~eiI~ j4i44t~.~4 -,‘-~z~4 4- +4’*v~iI ~j~~’-N

4A1,L, ~ ~ I ~ 1’~ ~~ -‘

1’1 4rt4 kt Pt—? ~. The,, Slfl~.C Iirs., K —4 ~o

we have ~ ~ Su ~ ~, .~ ~ CC But ~ is arbitrary.

Hence ?~ —~ ~ as ~ —~ ~. Q.E.D.

Theorem XXIX. (Abel’s Theorem)

- Ob

If the series ~ 4t~t’ 4, ~ 1,, 13, ~

converge where C~. s ~ ~Z - ~, ~ ~ ,~ A 13’, Rearrangements. Defijition, Let ~ be a sequence in which for each k one and only one

positive integer appears. i.e., ~ IV is the set of non—negative

integers in a different order. Another way of saying this is: 39 the series is~ I I function which maps the non—negative integers onto the non—negative intege*s.

Consider the series Y4,,,Set 4’~ ~l, ~, Then the series is called a rearrangement of ‘5~a~. In general

however, we cannot conclude necessarily that .6 converges then 00

q..~ Consider the series, ~. — and the series Zi5;1-::3 ~ii ~ I

_~i~ 4L ~ ..j.. - - -, 06servc ~Ii~f i~ ~ h) et~I 00

~-~“ (t — -t~ ÷i\~.i~(0 >S an€J ‘c s”~ ~j-_~_~h’I.,’3 .,. q.)q-~j_. _..~. V., I

then (i ~~45’ (ci~c £~3,-3 i q~..3 —~ >0 .ç:01.411 n)

Note that , ~%~4r3 1~~.~q-( á~1~ 7. -1- 4~ ~ rC~There fore we

have that 56 5 and hence the assertion made above.

Definition.

If every rearrangement of a series converges to the same

sum, then that series is said to be unconditionally convergent.

Thaorem Zfl. If the series ~ converges absolutely then the series ~

is unconditionally convergent.

Proof: (a) Suppose 1h 0 Consider the series

~. ‘~ as any rearrangement of ~ 4 ~. Set S ‘,~

€~ ~ ~ ~ Set- S~ a, +t?~ j-Q34----f4i,, Let

~ tIiA ~ •,. L1,~3, ~ * ~ f,.~44~Thus 5 ~ ~ £ ~ •~I =1

for every, Henc~*h ‘~‘ exists and is less than or equal to ~ 40

Call this 1imit~. Therefore 3’~ S Equivalently we can

consider ~ as a rearrangement of ~ Then following the same type of reasoning as was exhibited above we obtain 5 ~

Therefore $ t (b) Suppose ~ 4~ converges absolutely. Then ~ 14J

converges. Set p~ ~ ____ ~ o ~nd

— fo~ au ii. Moreover, the series~~and ~ converge. By part (a), any rearrangement of the series~P.,and ~ say, , and ~ converge to the same sums as and

~ respectively. i.e. IfIo:~pbandQ~≥.~,, ~ asd ~ Q. 6~14t r”~ ~ l~~÷4, 44i1d ~ 2I4I~l a,1)

t4iø+ ““‘~ I’~ *~hI4,I. ~f ~ P-~ aweI ≥—I~~~1 Zp~ ~ tf#Q. Eje~;vAI.es~ly ~

P-€~ 41f4 ~JAH P.,-~. IICNCq~ ~ anc) ~ ~ C~APTER III

INTEGRATION

Let f be a real valued function defined on the closed in. terval r~~) b3, Then the set of points, Xb) ~k’I) ~) A’s) - - -. X where ~ and ~1(~ 1, is called a net and is denoted by 2’J [xj., x~] is a sub interval of (a’,, bJ, We denote the length of

4 ~ & —A’s-v, wh~r~ L: 1~2~ .~,--. ~ The- ii~rr’~ c4 lit i~s d~c’ -t~ bc r,ta~ a~s, ~ IN1~e~iac 44’~, i~Lti~. Lct ~ C Ct~-~, x~J. Th~r ~ r ?~Ufll, _!!_ Th~, ~c’o’ti~ + b € ~ ≥~. .c (~i) 4 ~‘e ~ ~ = f(~j) C~—x.3

••••

Definition.

Let .t1 be an arbitrary net for the interval ~4) kJ, Then the fimctionf is said to be integrable, (in the Riemann Sense) if and only if; exists and is finite. This unit (if it exists) is cafled the definite integral of f~) over the inter~aJ. [4~ 1,3 and is denoted by

The definite integral is a special sum. It is an infinite sum. The equality abcve is a transformation of quantity into quality. A quantity of sums, on the right, builds up a sum which is represented on the left. The left hand side ~ and yet is not a sum.

Immediate consequences of this Definition. 42

~ — ~ f(X)4’K, & 6

2. ) -~(‘)ç)d~( =0.

Ex~1e. Suppose ~(r2 ~ X • Find ~ ~x ~.

Consider; ~ ~ Xi.., (KeK~,) 4,,4’ J” ~ K~ (i~ ~

~ ~ ~ ~‘~‘ ~~‘~-‘ (Y~-k’~-’)

~ ~ (‘~—x~-~)

: (x,~ - t- ,~i, t - - - ~ ~ “I ~ ), ç1~ i ii~

- ,k’~,,. TIy’cE~,r~t )4 /dX ~j’~g ~

Properties of the Riemann Integral,

Theorem I. If “‘n ~ ~‘; exists, then it is unique. (~~r)—.~ i~’ C Proof: Suppose ~ ~ xi j; ~r ~

Suppose I~ ~ Then set ~: ~ [i~I ~. By ou~ assumption, there exists

q S, ~ such that ~Z - I ~‘ ~ ~ IN~’ ~s and there exists a S~ S~&) such that l~(~)~2’~ ..J~44 (Or ~ L.el ~ [s., $,J. T~ 2~’ jr-~j I ~~

~ 1 ~f(ç )4 ~ -~+ ~ ~‘ lN1’~, wk,d, impI(~ ~ Th~rL’f~rf. 2~J. ~.E.D. Theorem II. If i~’ and are integrable on [‘Y~ L~J and if 1~(k) 4 Th

r4, bJ, fIi~r S ~ ~ S4 ~

Proof: Set I S ~ mnd 5~’ J~’a~ J.

Suppose r > J. ~ ~ ~ (i - ~ There exists a

~ such that ~ f~ç~)4 r~ — I ( ‘ ~ ~ Also, there 43

exists a ~ such that ~ C’~ài~ 31’ ~, ~ I.r4’≤. Lef £r p,;,i ~

Then ~ ~ Ji & T- ~ ~ ~ INI ~ ~. This is an inequality which is incompatible with f~() ~ on ~ hi. Q.E.D.

Theorem III. If ~ is integrable on 1i, L’Jand ic is any constant, / then ~ f is intograble on £~i, and IC ftc) ~‘i~’. bJ ic 4

Proof: Let ~iobe given. There exists & ≤ ~fr) such that

~Z ~ 5~i~&)rJ*J ‘j~\ ~) ~- iN)’ S~ Observe that,

— k — 5r~arj~ lid b ~ 4ar )J~) .‘≤ N’t,i~ ~ ,cfO~’dc K d.c. ~ 4 J4

Theorem IV. If t’ and are integrable on £4) b) then f ~4

is intograble on [‘k, I,),, A.h ci, {e6() ~ I (s’)) ck ‘~th 2~ ç ~ e -~

Proof: Set I 4c and J~ ~‘ ~ Then for

~ >~ there exists a ~, &) such that E (~) OX.~ ~ foe 1141’ ~ AJ~so, there exists a iSg S2~) such that j(çfr~foe

lArI4:≤2.. flewe_ I ~f~:)i’r~ —I t -3 I

4 ~f(~j~i~ — II ± J ~ —J-1.’~~e co~ j~r~ ~.

Theorem V. If C and are integrable on C4~bJ and ~k) ≥ fCc) on [c, t~7 except possibly at a finite number of points on r4, i,J, 6 then ~ ~ 5~f(x)dit. A

Proof: let I £f~*)d~C ~ Assumei~and set é ~ ~ (Z J). Then there exists a positive numberSso small that for any netjjof norm less than ~ and for any choice of points

xI, ~ ~‘!)• -. ,J(~, the following inequalities hold simultaneously;

~ 4 (~ .~ 6 I -6 ~ ~ F(~)~~ cL. t3~’- 44

this implies an inequality inconsistent with the assumed inequality

that f(x3~~c).

Theorem VI. If 4

~‘fC*-~c1x + ~fCxdr.

Proof: Let I j (tr)cIK 4ncJ J çfcc)€JK. ~ ~

~ exs~s a ~ £,&) suLh ~ ~ —1 ~ ~

~ IiVl~ £~. ~ ~ ‘~‘ ≤~E~ such Higt

I ~, ‘‘~~ ~I c4. ~ LrI ~ ‘i*4,~ ~ SL4 4(~) f(~)~. ~ ~ ~ ~

1N1’≤. c’~erve 4~tf 1~~C~)4~ (z÷~I ~ fr~+.~j~J~,xI _(~[÷3~)J

I ~ ~ -rJ+i ~- f(~1~)I1A’I-1) j~ ~4x.~-I(

•&[~~ f(~~)IL~ ... 3) ~ ~ (NI ~ £. Th~~≤

I~ f (or ~ lii€rt?~or.c.. f(t)Jv. 1tJ1_—~o 4~:I ~ i÷:~ ;~ fin,’f.c Suw, ~ S4 f(x)dr eysfs 4nd I.i. ?~ S41’ f(x)~i~c ÷ ~ f(i’)dx. ~ 45 Theorem VII, Any function continuous on a closed interval

is integrable there. Proof: Since ~‘(x) is continuous on the closed interval

tA, b3 then it is uniformly continuous there. Assume k’ >4.~

Then, given & ~& there exists a positive number £ such that

r’-.. r’~’l’.~ implies ~ — Letibe any net of

norm less than6, and, letQ~ andtL for £-~

be defined as the minimum and maximum values respectively of

on the (sub—interval) ~ ~ If Y~ and are points of ~ such that 4fr~)~ Ci and fr~) i

where 1: i1 ~i and if the step functions OZvand i’~r~ are defined to have the values ct~ and 7~ respectively for

a’.’ 44 where I f~ ~t, - - - - n and the values P(~4 ~) f~ r

X tli ..‘ he re I, ~, ~ ii, th~ O~) ~ ‘ ‘~-tr) and

SEi’o~_ Lrcx)]d*~ (sr. £ ~f(r~)

— ~ ~v i~ ~ (v11.r0)~± C~b-m) :~• Q.E.D.

Theorem VIII. If the function ~ is bounded and continuous almost everywhere on (~, bJ then F is integrable there.

Proof: Let ~‘ be continuous on fm, b2 except for a finite

number of discontinujtjes k at ‘(~ ~. In any net over r~, ~, ~ is a point of at most two sub—intervals. i.e. t Kc where XI is

a net point. The oscillation in these intervals is ~ I~l - ~ é ?lCt

LW1 of ,e~b ~.Their contribution to ~ S ~ ~ r(ii - ~) b~. L~# ~ ~‘ £ where 6~ is so small that ii ((i~4-,,,) ~ .~ ~4 tE 4 nd M. - )il~~, 46

in all sub-intervals in which f is continuous. Thus we have $ -

Theorem IX. A function defined and monotonic on a closed interval is inte~rable there.

Proof: Assume for definit~vi~sS that ftx is monotonically increasing on C~ and for a given net ti’ define the step func

tions ~‘~) ~ ói~) for ~ ~ L 1~ ~ ~ ii d ~ .f~. ~t1~ f’x~ cw i’(~ 4id 5~Li-~- d4()Jdx

~ —f~)J~x ~i.wi ~ L~(4~)-~~,)1 lNlcfrc’b) — -E&O1.

By a previous Theorem (#iv), it was shown that

(‘lv) - d~ ~ H.Lprt~ 5co’)JA 1NJ L!~’-~~• which shows that ~ 4X~ exists. i.e. it is finite since this limit is less than fNl~[’b)’1$(~~)1 which is a finite quality. Q.E.D.

Thsorem X. First moan value ThWW4for Integrals. If 4~ is continuous on a closed interval ~ kJ then there exists a point ~ t j1.i i~J such that ~fGddx ~

Proof: Let iii and S denote the minimum and maximum values of f(~) on iq,~ j,J Then consider the sum ~) 4 ~ + * i E Lr~) 4 V.,

Th~en I’ll ~ ~. .s whe,~ j ~,,, 3,. -- arid ki 4L~

~ (vi) ~ ~ ~ S

But ~ r~ (b- a). Therefore the sum ~ f i-- t 47 lies between m(b-A) and £Cba) Since by definition of a definite integral as a limit of a sum we have ~ ~ ~ s s-a) then ~w ~ ~ ~ S. By setting I•4 ~ we see that for ~ ~(~)ean take on the value M since vu e.N~ S

~. z~ Ed lies between the minimum value of f(K) and the ma~iinum value of PC~) 0~f’.,bi Therefore f(~)(bt~) ~ Q,E.D.

Definition. f(r)~ ‘~ (K) almost everywhere on L”~ bJ if and only if (1) for points on j~ i~J which have length zero.

Definition. ~ means that a condition holds almost everywhere in (~,&1 if and only if it fails to hold on a set of points in (4jbJ of length zero. Proof: Let S be the set in question. Then, since S is countable, it can be enumerated in the following manner:

Let~>o begivenandlet

Note, (~ ~.V & ~ is an open covering for the set ~. i(s) ~ ‘(2~ ~)

~ ~ (Gj~ ~ Thereforef( $ )‘~t. But ~ is arbitrary there fore i(s) ô. QE.D.

Theorem XI. If a function is constant on L’,bJ and l.a particular j~ f&) t on f4,!,], tuCK? 5aeXdr~ k(b-t~j.

Proof: Observe that for any.W; 2 f(ç~) 4 V.~ = c~ 4 K~

K ~ K (ki-Xb iX2~X ~

~ x C~-~1 —~ 4$ Sc(x ~dx k(b-4). ~.E.D. 48

The Riemann — Stielt~es Integral. Definition. Let ~ be bounded on the closed interval t4~ 1,1. Let N be any~ net for the interval Let 4i’e

Let M~ I L4J~ ~‘6() and i~’~ I L, ~(z)~,i~e K 4.-Ct. Set

U. (c, ~,) ~ iVf~ ~ a 0 ~d L Cc, N) •~ 4~€~ 4 t~

cCr)dr ~ I 1 t. (jASf) ~ i~s~s,ann Luwev~ ft~fe~aI irdSfcWx i~b i~it) ~Riemann Upper Integral.

If; ~ then f is Riemann Integrablo on L’ b]~ i.e. f f the class of Riemann on bJ and ~ fdx

6 4 5 Riemana Integrable on f~ b2. A necessary condition for the existence of the Rieinann is

boundne~s and continuity almost everywhere. The Riemann integral

is a special case of the Riemann — Stieltjes integral.

Definition — Let ~( be nionotonic increasing on (4~, bJ. Then since

oL(a) ando€( b) and finite values, v~ is bounded on ra, ~1

Set 4 at~ (°~ -~‘~,) ~o, Co~ - - - . S~f L(4~.V~)

~ 1.t~ ~ wh’er’€- 4. ~ :, .~ .... Se.t U~f,N~.) ~ 1v1~ ~cc:

where ~. I, ~. .. Then -f~ dcc ~ I & LL (4~ J~i, ~) 4 R S upper

integral, and 5 4~ do~ I ~ I, I (~, .ht, °~ - ~ lower integral. 6 If ~qfdC( ~ 5 fcI~ thenf is Riemann — Stielt3es integrable to on Cii, I~i. Its common value is denoted by S~~’ ~ ~ ) d0L.

Let °~ be defined by S~ (~ ) ~ on L4, ~J. In this case, the Rieinann

integral reduces to the Riemann — Stieltjes Integral. Hence 4.€~ 49

= ~~‘j hence the Riemann integral is a special case of the

Rienaann — Stieltjes Integral.

Notations ~ I ~I, 4~(x), gvh.Lvc 4’~_~ £ X 4f~~ ~ ~Ib f(~~), wIieec X’t-g 4 *~~Xi

~ p44: ~

~ s~t LA(c,i~r) :t (%-j~~4g~•,

LLc,.w)~ :..“~ ~

We assume f is bounded on L4, &J. Then there exists an fri **4 ~ such that 1i ~ ~≤t)~ M. If ~~k) ‘~‘~ exists, then s~, (i- a)

~ kt(~6).Moreover ~ = (i-~t) ‘ j (c,.V) ≤

~ M~ 4L4 (~1&-a).From these inequalities, for a bounded function cdx and al~~ays exists. Consider the diagram, where A~.

Am~ Ac ~

~ Ac~ ~ ______— 4’,’ There fore

Oi(b~) ~ ~. 1Vl(1~). Figure 3

Definition. k*is a refinement of the netN if and on1Y if

Suppose ..W1 and .1412. are two nets for [4) kJ~ Then .W * is the common refinement of N, and j~4~ if and only’ if j~ft .N~ V N,.. 50

Theorem XII. If is a refinement of TthenL(f,N~)~E1(f,1~2) ~u(~N~c)~ u~Cc,Nr~).

Proof: Suppose we consider the i ~a~4yT Lc~.Nac) ~ I-(ca~r~),

and suppose N~ contains just one point more than ~, ~‘t ~

for some -~ where )s~-~ and ~ are two successive points of ~-

Let ~1=~tI~ o~ ~ X~-i~X~ $~ ~ w ~ o~ f(x)~L%e.sc X4~ ~‘-‘X~•

From this we haverni~uL a~c~w2.l4es~~ ~ ~

Observe that 1_(~, bJ~o~)— LG,N,~)~ UJe ~ wz (~x~ _X~o&)

If has 4c points more than g we repeat the above argument

%L times. HenceL(~’~) ~ L(f,x~r~o~) . In a similar manner we prove LL(~, ~7c~.) Li(4,.b3,oL) Theorem XIII.

Proof: Let j~Ii and j~ be nets for the int~rva1 taN]’ By Theorem XI, if is a common refinement of N1 and .P~a then

~~ L1t’!’.L) ~ LL(c,N,~L~ 1l”ec L(f,git) s u.(4~,w, ~i). Fa~ bla.

Then the L~ia (of all ~ )gives (L~a~) ~ ~+ i ‘~(L ir.,1at).

Observe that ~ ~Ur v{f3~~oc) ~isen. ~ ii~cc .~ j~ (ç,i~ ,oL) ~

-~

Sian ci~r~ *

Definition. means 4: is Riemann — Stieltjes integrable on ~1i1, i.e~ is Riemann integrable with respect toOC on (.whereo~. is monotone increasing on

Theorem XIV. .~. &1U~c) if and only if for € >0 there exists a

net N such that i*(f~ N1~) L(c3 N~c~)~ £. Proof: (1) Assume for any ~>o there exists a net N

such that (u~-~4 £. ~ Uh(f,.~L) ~ SL.I~L ~. ~ ~ ~t,iL). 51

implies that L4~c~ ~4a1~ ~ U(cc~).(c.N,~)o Hence

for the E and l~T in our assumption we have ) ~ - ~ ad~. 46

Therefore since ~ is arbitrary, we have i ~ f~R(cL).

Proo1:(~) Assume -~ K(o~) on ~a, ~1 and let £‘~ ~ be given.

Then there exists nets t~r1 and ~ such that u(fg11aL) —~ ~ ~

~ V

~ ~as. ~ since ~ is Riemann — Sti€lt1et

Integrable. Let $1 be a common refinement of N’~ and $‘a • Then ~ ~a,DL) ~id ~ . From above, ~‘ça.~ ~~

~~ ~ ~ ~~ V Definition. ~i. ~ u)L~ ~ i~r, ~L~i,gFLL jJ~)

~ ‘~. Pic%14X ai.~ ~ —

Theorem XV. If ç is continuous on ~ then 4~(~) ~ Proof: Let C~c be given in an arbitrary manner. Let ~

be such that ~tO’3~ (~ivt ~ 6 Then ç continuous on V implies f is uniformly continuous on ~ Hence, there exists a

~ (Er) such that 1c~s - -foc~) ~ ~ when j:(’ - ~ Let ~

be so small that ~ ~ 4 ‘,p • Then observe; ~ L(j,b1o~) = ‘t~ ±4cC

~ ~-w~ ~ .~T . ~n.s$c~ ~~Ps(pL) t~?i~

The Riemann - S*~i4js Integral as a limit of a sum.

Let ~ ~t)C~-~1 ~ ~~-i- s (ç ,~ ~i~) c (ç~) ~

Then consider S V If this limit exists call it 1... 52

We will show later that L~

Note — ~ is assumed to be bounded and cL is assumed to be monotone increasing.

Theorem XVI. If ~ (~) and if ø( is continuous, then ~ ~ibM S (ç, i~t,o~) = ~ ~ ~W. Convereley if ~ ~ holds, then Proof: (i): Let E~6 be given arbitrarily. There exists Pb

~T(r.) such thatU(f,IT.OL) ~ ‘ Let

Take Zj~ where ~ is the number of intervalç into which E~3~ is divided byN’ Letg be any net for which~f’6 Consider the sum S(c~ 11 ~ Each interval of n which contains ii 6 a point of N contributes no more than~- A(N)t’1(t~—~ € ~

to L&(f,N,cL~. Thus r~i 4 ‘ ~ .~.

for ~N I’ ~ In a similar manner we can find & such that if

~hTk ~ then ~ ~ ‘ Let *“s ~

Then S(c,iJ,~L~ ~ ~ ~d ~(ç,I,0L) ~ ~ Hence fo~ ~1 it~r~&.L.e., \~ S(c,~~r~) =

Proof: (ii): Assume ~ 5 (f,14bL) = lgfr’b

Then for any €~.cs there exists & (E~) such that L ~ ‘ 5(No~) <

lirI~ S. ~ 4tj: ~txA~) xñ Then for a fixed VL taking and

~ ~- %~ L cc.~1~) ~ ~ ~ ,~ t*~tlI~$

L(f,kFgo~fl ~ ~‘ ~ 1N~L ~.

Properties of the Riemann — Stièltj~s Integral. S3

1. If f and ? (~) on ~ ~ and 6Cx ~ ~ cs)~ 4Lc..~i

\~SC~b4 ~

2. If ~ ~a~3≥’ tE-~L~)WX)~c\%eb, ~ ~

b~6 tc~,W~ ~ ~ ~~

3 ~ ~i ç(~~ ~

~ b~4 ‘~)V~ ~ ,e

it. If ~. on t~ ~ and Q is any non-zero constant,

then ~t ~~(5 oC~ ‘a~

S. If ~ \~p~ t.a~1 ~

~ ~

Proof:; We wish to show that ~\ t on . Iaet be given arbitrari)~. Then there exists an ~ such that

‘... (c, ~ ~ ~ ~ (i)\~ Os)- ~c~’p1~ \\~s~jc~\\s

Iaet~. ~ where ~ ~ ~ XL ~ ~- = --

Let ~i ~ where X ~ ~ ~ )(~ a~ \, - - Observe that;~AC~f\i b ~ 4xL I

~ ~ ~ ~ hence ~~~b(c(’~. Q.E.D.

~— and ~ - where M~ x (e, f) Proof (ii). If ~ 2

~~ \f~=-ç44ç~bli .,~

Observe thatc4 and ~ on ~ Now observe that\~~0L\ \c~i(f1~ ≥~~\ ~ - ~Vc~cL\ C. ~ ‘c~eI\ Si’

~ S~3.0L

Therefore \~\ ~

Theorem XVII. ~ b ~ ~

Proof: It is sufficient to show that t. ~ (~L~ ~ in~lies ~ -

Let t~ ~ \~c,c)\ s~ ~ ~, \~(~\ €

Let E.’>C~ be given arbitrarUy and let ~ be such that

\~ ~\ ‘ on V~ )W~. Then there exists an i~ such that

~. • Observe that; ~1~Y,~t) 00

~~ ~ (~_,iOt~oLi ~~(t1~L’rmL)

~~

~ s;23bc0L) ~‘(_~1V~ ~

Scope of Stieltjes’ Integration. Definition. Unit step function. ~; — j~_~ ~>~‘

Theorem XVIII. If c is continuous on EaV~ and if for ~

~

Proof: Let IT be any net with C~ as a point of subdivision,

then observe that; S (S,}~, OL) ~ ç(~~.~’) ã°~. . ~ (c)

~ ~c~) ~ ~ s (ç, .N; ~) cCC). L~.) ~ ci ic). (~p•

Note - If CL ~ the theorem does not hold.

If ~ (X) is constant, then ~ clot o since /~ b& Cl ~ = C~

Theorem fli. If (~ ‘2 c~ and ‘~2 ~ converges let be ar~r sequence of distinct points in [~a ~ . and let °‘ (IC)

~c i(x- X*~’). Then i~ -ç is continuous on ~ ~

Proof: Observe that since (~ I (x —xw) ≤ , the series

~‘. (.x’) Zc~~ I (~- x~ converges by the con~arison test. ie.

Q ~!. (~c-x~~) ~ C~ for afl h~ • The function ~ is evident]~ mor~otonic, therefore let E.’ ~ be given arbitrari3y. Then there exists an 1& ~- N (5’~) such that ~ Q’m ~ °~

Note - ~L(~=c~ awc~ eLC.!p):: ~ Do Set~(X)v~~(X~!’) i

= (~ LCjc—~ti).

= EL

Then ~ 4 ~

Since is continuous on V ~~ol t~iere exists ‘a” I~l ‘ ~ such that ~ o~’ b ~ Now observe that ~ cI bC ~

2 L (x-x1~ ~u~c ~ c c~)

Hence\~oL_i~ ~ \S~\*~2\~ ~ 56

‘. . E::. ~

~. — ~ ~ ~‘zc~

Theorem )CL~. If on t3~V~ and if cL()~) i~ differentiable onX~V~ then

=

Proof: Let ~ be a net fort~1~. Then letj~ ~ be chosen so that ~ (~) - ~ ~) (~ x~1).

(13y virtue of the Mean Value Theorem )~. &w, observe that

~c~~tL ~~ I—’

Hence if t~ ‘Owe have cL’c~c) ~. Q.E.O.

The Fundamental Theorem of Calculus.

Theorem XXI. Let f be Riemann integrable on Set t:(LC) ~ ~k~) ~ Then ~ 00 is continuous o~

If ~ is continuous at X. ~ then ~‘(jL)

Theorem ~(XI( If ~t1 on ~ and if ~‘(jO ~($) i.e. ~ (,C) is a primitive of ~ (ac) then

C3L~ 4≥%;x ca~~ — c (p).

Both of the above theorems are considered to be fundamental theorems of Calculus.

Proof I; Let~(e~’~.—- ~o~x j U3LVLI. I~XI~04 ax Thea1,byTheoremE~c(~bio’°~rc4~c&4 ~c4a~c

— \ ~.C~)à~ir ~ \~cc~) By Theorem Z,

for some number ‘~ , where )(‘ ~ < X~ 4 ~ Therefore if ~ X ~- 0

~ ~ ~ _ hence the theorem.

Proof II; By Theorem I and the hypothesis of Theorem II, the two functions ~ ~(4) ~ 4 and 1 (10 have the same derivative on ~‘~a Therefore, (by Theorem II, Consequence of N.y. P.), they differ by a constant, and Q) S: ç (;c) ~ — ç ~).~ Q • V ~, then, substituting in (i) we get ~: ~ l≥L4 ~ ‘F c~ — ~ for the particular value )( ~ is the desired result and hence the theorem.

Bounded Variation.

Definition. Let II be ar~y net for ~ Let 4 be defined on

• Then, ~ o4 ~ ~iç~çj) - finite, then ~ is said to be of bounded variation over

Examples: 1. Monotonic Function.

Let ~ be monotone increasing on ~J

Then ~‘C~ ~ \~~) — N = 1ch)—-i~(~) <4ooj 44e~ V(f)

is of Bounded Variation on

a. Ax~y function .f which has a bounded derivative on ~ is of bounded ~iariation there.

Proof: By assun~tioñ there exists a I~° such that

V(ç~.. ~ c~—j on ~ ~ Note that t~’~ ~“ ‘~ ~ ~ (~) ~(JC&.)

~ ~ f ~~ ~ ~ 4.0

3. A function may be continuous on t~a ~‘] and ar~r net not be of Bounded variation there. Consider ~ , defined by; 3 e~ ~

1 ç .2_ ~ . ~ 2 V.. ~ -‘ S

Then y(ç) ~, ~ 1coc~)- foe~t~(.

4. -- --“ ~ ijj~~) ~-

, . ‘~- -~ i.. 4 -~ ~ - - - 4 -k -~ ~- ~b ~ —~ 4- O~ ~i t e • Hence 4: cannot be of bounded variation on

Theorem ~U. A function of bounded variation on ~,h1 is bounded.

Proof: Let V(*~ I’~L, and let hT be a net for ~ where

X<~o.Then observe that~)C~)~ j’~(XA.) ~ ~

~ ~~(‘b~’ = ~ ~ Vca~ ~1e~ ç~ ~1 4(f~~Vo_f(~)i-f?~ ~

Therefore 4~ is bounded on t~, h~.

Theorem Xi~iV. If 4~ and tare of bounded variation on ~ W] then .f ± and ~. are of bounded variation on ~, 11 ‘WI Proof (i). Let 14~ be aw net for ~ Then~ I

± ~ — ‘~ ~

- sic-i)) I ~ ~ ~ ~~-‘)I ± 1 ~ d.. I ~V(f’) ~

—

Therefore ~j ~ j. ~ v(c ~ v c~ Hence 4: is of t -a, hi t~,b~1 bounded ~ariation on t~ ,~• (ii) Let ~4 and ~ be positive constants such that

l~(a~~4 ~$~‘ ~ ~ Observe that~L)~&~)

— çC)(~I~ )(~*(~\ ~ ~L) -g~~ )4c~c~-~~ ~ ~,(ic~) fc~ci-o —

.~ — ~ ~ lf~.~-o fl~ ~ç1u-t~) ~ ~ - 4~(ICL-I)l.

~ V(c) ~ ~~:L4I~.4. ~ E 4. A V(). ~

Therefore J~~1S is of bounded variation on Q.E.D.

Corofla~y-.

If -~ and ~,are monotone increasing on ta)~i:I then 4~- ~ is of bounded variation on l~, hi.

Definition. Let ~ be of bounded variation on t~ah~ Then, for Xtt~~ let 11 be an arbitrary net for t’~,i~ 1. Set ~4 L

= ~, X~, ~1 ~

- < X~ X~o. Split the set of integers into two

classes A and b accordingly as is positive or negative respectively. Then we define ~) l~ ~

~ ~‘~‘ ~ are cafled the positive, negative and 60

total variation function of 4! on h~ respective)~. Observe that

~(~ N (.a’) (~. Also~ note that ~ t~.(ic] ~~~-X)V (~ implies

t4&i~) and ~Cjc) are finite in ta, ~\ since ‘J(~) ~ V (~) —

Theorem X$V

If f is of bounded variation on ~ ~. then there exists:. monotone increasing functionsV, ~ and V such that (i)-cc~ (ii) ~(~)~c3o+.gc’~’’ Proof ~ (l)Let )T be an arbitrary net for’)) xl. Then observe

that cC~L) — cçc.~ ~cCL.~) L.~-’~ Q.) ~ ~ ~

so that, ~ ~ ~ 3c ~OO- \C~)

~- • Cs i~4 çooc~)+~)O~, which in turn implies that SC.~ ~ 1k.~C) —~YOc)~

Gail this inequality. (2), Then (1) can be stated as ~ ~—. ~C.$~)

‘t ~) ~t~L which implies ~ 00 f ~. Lt~) This implies that ~(jc) — S. ~ ~ $~. ~ ~t (~O ~ which in turn implies

that ~. - ~ ~ ? (~) — ~4 (iC~. CaU this inequality (3).

Therefore by (2) and (3) ç(’L — ~ ~ ~

(ii)\)($~ ‘~6c) i- N~O. t~A~ ~k,a~r c(10 — =~L~r- c.~-~31 ~ ~ c~) ~

A ~‘- ~ ~* ~ TCs.)4- .t~rQO, which implies ~ ~ ~ (~cJ. Now, adding (k) and (5) we

have that, .2. ç L. ç Oc.) - çL~) ~~ ~i4~ L 61

~ Cs) - ~‘~- ~ which in turn implies that 2- ~ (2c) ~ ~ Cjc)

~. (~) ~- ‘.J (iO. By part one of the theorem (.~) ~C~) ~ C~O N (ia

Hence ~. ‘~c~ ~ ‘~Oc) - 1~C$~ # ‘J (iO ~ ~ ~ ,~i (ic) ~. — L~’ (c) ~-‘J (ic) implies that ?GO ~- ~(.ic3 4- ~4 (J0 Therefore ~C~Q i-bfli) ‘.1 (ic).

CoroUa~r. If ~ (s~ is of bounded variation on ~ then c exists for ar~ ~~ and çOc~) exists for ax~ ~(.t (a1cX’

Moreover, the set of points ~t ~a, ~\ where ~ is dis continuous is at most denumenable,.

Theorem

If ç is of bounded variation on ~j WI and if there exist functions ?~ and ~t ~ on ~N1 such that (0 ~Cit) - S~’) ?% (~O ~T’~ Cs) V ($~) ≥. vu~~ ~ V L?~ ~ ‘I Cs), where? and t~ are positive and negative variation functions of f~ on ~ and represents the total variation over La )~1 Proof: Since the addition of a constant to a function does not change its total variation, we may asswae that ~ (a) 1’~1 ~(~) = 0. Then~ ~O3~; _&c~ From (i) and Theorem ‘~X iv we have that, ~ But on the other hand ~ (~ ~ Hence (ii) ~k..i,) I

~≥u~ ~~ ~ ç~-~c~ i~o~ ~~-aoo~ ~(~-t’f~(iQ,

~(g~u) ~~-1~%~)’ Adding (ii) and (iii) we have that ~ (S)

&,,e., ~ ~ Subtracting (ii) and (iii) we have that 6a

~ (~. Lee, ~ ~

Generalization of the Riemann — Stieltges Integral. Definition.

Let -~ and be defined on ~,i. Let J’~V be az~ net for

Then 3~ is said to be integrable with respect to ~, over ~ if and only if ~ (~N-~ ~) exists and is finite. If this limit exists and is finite, then we say ~ (~) a~

~cc,~L&r ~~

Remarks. If f is monotone increasing then ~s~(~) ~ If is differentiable on (~a~) and ~‘is continuous on ~ then

Properties of the Riemann - Stieltjes Integral.

(a) If~~L~4(~ ‘~ t~’t~1V~1 V~i~’ ~ ~ ~ ~ (b) If çj~ ~ ~ ~ \~

Cc)

Cd) If ~ e~ ~ ç t any constant ~ then ~

(e) If is constant ont~1~ then every function ~ -~

3 63

(f) I~ ~ ~ ~ ~ ~ ~ ~

AU of the above properties foflow from the definition of ~ (~) -

Notations: 1. ~ if and only if S (N, 4:) exists and corresponds to

2. ~ where ØCis monotone increasing, if and only if exists and corresponds to•

3. ~ where is not necessarily monotone increasing if and only if ~ (k 1, ~) exists and corresponds to

Theorem ~S,)W1I.

If ~ ~ ~ ~ ‘I~ 1~

S~ ~) c Proof: LetN be aw net for \~, h~ In particular, let ~

be the set of points i~ [‘ei X0 ,3~ ?C2, - - ~ - - ~ Xn.

Let! ~ “-‘-~ g ~ ~Y&~T~ “L~Z’ “!~+. where ~, < ~- ‘- — — - • Observe that X L- ~

where t ~‘ -- - - ~‘ ~ By ~ partial summation fox~uula

above ~ (II, ~, 3~) ~ ~ èj~) [~ocL) ~ ()C*.-L)] — ~ ~

— ~ça) ~(~) ~~ ‘ (a’) 61~

Now letting~IV~C3 implies ~“c and we have that

-~ ~ ~ 1’) ~ ~) ~i -+ ~ ~ ~

Hence the theorem.

CorollaV!y 1.

If ~‘ and bt are continuous on ~ then

~ ~ cG~)~c~- ~ ~

Coroflaf~r 2. If c is of bounded ~ariation on ~1,) and if is continuous on t~~S then c ~ t~

Proof: It is sufficient to show that ~ ~

Since ~ is of bounded variation then ~ - ~ I~’6’)

&.L., S ~- ~ ~e~i ~c~c i

— ~ ~- ~ Theorem ~ If ~ and 4 are continuous on ~1 and if c~≥is strict]~ increasing on ~ ,V\ and ‘44 is the inverse function of 4~ then

~ (v tv)’) ~ L\ ~ where we assume cj~ (jC)~

Proofs LetJ!t be the net where a Xb~ X~< ~---- ~-X~i\’

Note that ~ 4)j (~) where ~ 0, 1, 2,3s • Now consider the

netj~ [4~~ ‘i~ ‘(i, Y~i~ 11 whore

‘?‘o~ y1 ~. L ~f~.Then observe that for ~ ~X~-.i ,,XL’l whereJ_~’1~~r -.“, ‘~c(yLXX~-XL~ ~ 6~

Now, letting Lj~ii -~ C we have that tbfL —‘ 0 by uniform continuity of

4) We have also that LMI —‘ o ~ (~ -

L~ ~-c ~i’c~ [‘? (~) — ‘1’ ~ ~ c ~r1 -~t~ ~l. (~L~) a1,

Theorem ~ First Mean Value Theorem.

If ç is continuous and ~. is monotonic on i~, 1,’) then there exists a point ~.1’-~ such that ~a.L

Proof: I~et ~ be the ~ of ~ (~) where ~c. E ~

~a ~ ~ c(~>. ~ ~ ≥~L.

~ ‘S.f’ ~ ~ Hence, there exist a number ~ such that ~

But since f is continuous on ~ there exists a point X f. ~ such that ~ ~ ~ OO~ Therefore ~-ç~x0~ c(≥C) t~~L(1~- Y\. Q.E.D.

Remarks. It may not be possible to select a point ~

‘ ~‘ ‘f ~ ~ Observe that for consider o~’C) 4 ~

~ )Cv~ ~ S0~1~1)~vL(XJ) -.

=~te~ ~

Theorem XE~(. Second Mean Value Theorem. If ~ is monotonie and is continuous, then there exists a point )( ~ ‘b\ such that c~’~ \p~c?() -

— By Theoremwi. we have ~ S 66

~ f ~) ~ ~ (a) ~ (4) S4 ~ ~ f. Now by the first n~an vab~e

Theorem there exist a point ~ ~ fti, t1 such that

~(x) [~)~4(6L)].

5 d ~ c&~) [g — + c(s) [~ (b) — ~ L~ BIBLIOGRAPHY

Courant, R. Differential and Integral Calculus. Translated by McShane E. J. 2nd. ad. Vol I. New York: Nordeman Pub— lishing Company, Inc. 1942

Granville, William Anthony, Smith, Percey F., and Longley, William Raymond. Elements of Calculus. Boston: Giun and Company 1946.

Olmated, John N. H. Intermediate Analysis. New York: Appleton— Century — Crofta, Inc., 1956

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