DOCSLIB.ORG

Infinite Series: Definitions and Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Infinite Series: Definitions and Theorems Math 347 Infinite Series: Definitions and Theorems A.J. Hildebrand Infinite Series: Definitions and Theorems Definitions P1 • Infinite series: An infinite series is a formal expression of the form k=1 ak, i.e., a sum over (countably) infinitely many real numbers.1 P1 Pn • Partial sums: The partial sums of a series k=1 ak, are defined as sn = k=1 ak, i.e., sn is the sum of the first n terms of the series. P1 • Convergence and divergence: Convergence of an infinite series k=1 ak is defined in terms of convergence of the sequence of its partial sums: 1 n X X ak converges () lim ak exists () fsng converges n!1 k=1 k=1 P1 The series k=1 ak is said to be divergent if it does not converge in the above sense, i.e., if the sequence of partial sums fsng does not converge. For a convergent series, its sum is defined as the limit of its partial sums: n X s = lim sn = lim ak n!1 n!1 k=1 P1 P1 • Absolute convergence: A series k=1 ak is said to be absolutely convergent if k=1 jakj con- verges. Theorems The following are some standard results about series that you should know. Except for the Alternating Series Test (whose proof is a bit tricky), all of these results can easily be proved using the above definition of convergence of series and results about infinite sequences (e.g, algebraic properties, Monotone Convergence Theorem, and Cauchy Criterion). Try to carefully work out each of these proofs. This is a great exercise in applying the definitions and theorems about sequences and series. P1 • Cauchy Criterion for infinite series: An infinite series k=1 ak converges if and only if it satisfies the Cauchy Criterion for infinite series: n X For every > 0 there exists N 2 N such that for any integers n > m ≥ N, ak < k=m+1 P1 • n-th term test: If k=1 ak converges, then limn!1 an = 0. • Algebraic properties: P1 P1 { Sum Property: If k=1 ak and k=1 bk converge with sums A and B respectively, then so does P1 k=1(ak + bk), with sum A + B. P1 P1 { Scaling: If k=1 ak converges with sum A, and c 2 R, then k=1(cak) converges, with sum cA. 1 We consider only series in which the terms ak are real numbers. 1 Math 347 Infinite Series: Definitions and Theorems A.J. Hildebrand P1 P1 • Comparison test: If jakj ≤ bk for all k 2 N and k=1 bk converges, then k=1 ak converges as well. More generally, if there exist constants C 2 R and N 2 N such that jakj ≤ Cbk holds for all k ≥ N, P1 P1 then the convergence of k=1 bk implies that of k=1 ak. • Absolute convergence implies convergence: An absolutely convergent series is convergent. 2 • Alternating series test: If the terms ak (i) alternate in sign (i.e., are negative for odd k and positive for even k, or vice versa), (ii) are nonincreasing in absolute value (i.e., satisfy jak+1j ≤ jakj P1 for all k 2 N), and (iii) have limit 0 as k ! 1, then the series k=1 ak converges. Two Classical Series The following are important examples of series that you should be familiar with. You should know how to prove the convergence resp. divergence of these series. P1 k • Geometric series with ratio r: k=0 r . This series converges if jrj < 1, with sum 1=(1 − r), and diverges if jrj ≥ 1. P1 1 • Harmonic series: k=1 k . This series diverges. Series versus Sequences By far the most common mistake in dealing with infinite series is to mix up a series with a sequence, and P1 apply theorems and properties of sequences fang to series n=1 an, or vice versa. A sequence is simply an infinite list of terms, a1; a2;::: , while a series is an infinite sum of terms, P1 a1 +a2 +a3 +··· , or n=1 an; Convergence of a series is defined in terms of the sequence of the corresponding partial sums sn = a1 + a2 + ··· + an; it is not equivalent to the convergence of the sequence of its terms fang. Here are some examples illustrating the difference. P1 • The sequence f1=ng converges to 0, whereas the series n=1 1=n is divergent. n P1 n • The sequence f1=2 g converges with limit 0, whereas the series n=0 1=2 converges, with the sum equal to 1=(1 − 1=2) = 2 (by the geometric series formula). • The product of two convergent sequences fang and fbng, i.e., the sequence fanbng, converges, but P1 P1 the analogous result does not hold for series: The convergence of the series n=1 an and n=1 bn P1 does not imply that of the series n=1 anbn. (For a counterexample consider an alternating series P1 n p such as n=1(−1) = n.) P1 • If the series n=1 an converges, then the sequence fang converges to 0 (by the n-th term test). The converse, however, is not true. (A counterexample is the harmonic series.) 2As mentioned, the proof of this result is somewhat tricky; you need to know the result, but not the proof. 2.
Load more

Recommended publications
  • Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited)
    Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited) Limits as ‘ x ’ Approaches Infinity At times you’ll need to know the behavior of a function or an expression as the inputs get increasingly larger … larger in the positive and negative directions. We can evaluate this using the limit limf ( x ) and limf ( x ) . x→ ∞ x→ −∞ Obviously, you cannot use direct substitution when it comes to these limits. Infinity is not a number, but a way of denoting how the inputs for a function can grow without any bound. You see limits for x approaching infinity used a lot with fractional functions. 1 Ex) Evaluate lim using a graph. x→ ∞ x A more general version of this limit which will help us out in the long run is this … GENERALIZATION For any expression (or function) in the form CONSTANT , this limit is always true POWER OF X CONSTANT lim = x→ ∞ xn HOW TO EVALUATE A LIMIT AT INFINITY FOR A RATIONAL FUNCTION Step 1: Take the highest power of x in the function’s denominator and divide each term of the fraction by this x power. Step 2: Apply the limit to each term in both numerator and denominator and remember: n limC / x = 0 and lim C= C where ‘C’ is a constant. x→ ∞ x→ ∞ Step 3: Carefully analyze the results to see if the answer is either a finite number or ‘ ∞ ’ or ‘ − ∞ ’ 6x − 3 Ex) Evaluate the limit lim . x→ ∞ 5+ 2 x 3− 2x − 5 x 2 Ex) Evaluate the limit lim . x→ ∞ 2x + 7 5x+ 2 x −2 Ex) Evaluate the limit lim .
    [Show full text]
  • Section 8.8: Improper Integrals
    Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up.
    [Show full text]
  • Ch. 15 Power Series, Taylor Series
    Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex.
    [Show full text]
  • Arxiv:1207.1472V2 [Math.CV]
    SOME SIMPLIFICATIONS IN THE PRESENTATIONS OF COMPLEX POWER SERIES AND UNORDERED SUMS OSWALDO RIO BRANCO DE OLIVEIRA Abstract. This text provides very easy and short proofs of some basic prop- erties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor’s series, Prin- ciple of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integra- tion theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a mininum. Mathematics Subject Classification: 30B10, 40B05, 40C15, 40-01, 97I30, 97I80 Key words and phrases: Power Series, Multiple Sequences, Series, Summability, Complex Analysis, Functions of a Complex Variable. Contents 1. Introduction 1 2. Preliminaries 2 3. Absolutely Convergent Series and Commutativity 3 4. Unordered Countable Sums and Commutativity 5 5. Unordered Countable Sums and Associativity. 9 6. Sum of a Double Sequence and The Cauchy Product 10 7. Power Series - Algebraic Properties 11 8. Power Series - Analytic Properties 14 References 17 arXiv:1207.1472v2 [math.CV] 27 Jul 2012 1. Introduction The objective of this work is to provide a simplification of the theory of un- ordered sums of a family of complex numbers (in particular, for a countable family of complex numbers) as well as very easy proofs of basic operations and properties concerning complex power series, such as addition, scalar multiplication, multipli- cation, division, rearrangement, composition, differentiation (see Apostol [2] and Vyborny [21]), Taylor’s formula, principle of isolated zeros, uniqueness, principle of identity, and binomial series.
    [Show full text]
  • INFINITE SERIES in P-ADIC FIELDS
    INFINITE SERIES IN p-ADIC FIELDS KEITH CONRAD 1. Introduction One of the major topics in a course on real analysis is the representation of functions as power series X n anx ; n≥0 where the coefficients an are real numbers and the variable x belongs to R. In these notes we will develop the theory of power series over complete nonarchimedean fields. Let K be a field that is complete with respect to a nontrivial nonarchimedean absolute value j · j, such as Qp with its absolute value j · jp. We will look at power series over K (that is, with coefficients in K) and a variable x coming from K. Such series share some properties in common with real power series: (1) There is a radius of convergence R (a nonnegative real number, or 1), for which there is convergence at x 2 K when jxj < R and not when jxj > R. (2) A power series is uniformly continuous on each closed disc (of positive radius) where it converges. (3) A power series can be differentiated termwise in its disc of convergence. There are also some contrasts: (1) The convergence of a power series at its radius of convergence R does not exhibit mixed behavior: it is either convergent at all x with jxj = R or not convergent at P n all x with jxj = R, unlike n≥1 x =n on R, where R = 1 and there is convergence at x = −1 but not at x = 1. (2) A power series can be expanded around a new point in its disc of convergence and the new series has exactly the same disc of convergence as the original series.
    [Show full text]
  • 13 Limits and the Foundations of Calculus
    13 Limits and the Foundations of Calculus We have· developed some of the basic theorems in calculus without reference to limits. However limits are very important in mathematics and cannot be ignored. They are crucial for topics such as infmite series, improper integrals, and multi­ variable calculus. In this last section we shall prove that our approach to calculus is equivalent to the usual approach via limits. (The going will be easier if you review the basic properties of limits from your standard calculus text, but we shall neither prove nor use the limit theorems.) Limits and Continuity Let {be a function defined on some open interval containing xo, except possibly at Xo itself, and let 1 be a real number. There are two defmitions of the· state­ ment lim{(x) = 1 x-+xo Condition 1 1. Given any number CI < l, there is an interval (al> b l ) containing Xo such that CI <{(x) ifal <x < b i and x ;6xo. 2. Given any number Cz > I, there is an interval (a2, b2) containing Xo such that Cz > [(x) ifa2 <x< b 2 and x :;Cxo. Condition 2 Given any positive number €, there is a positive number 0 such that If(x) -11 < € whenever Ix - x 0 I< 5 and x ;6 x o. Depending upon circumstances, one or the other of these conditions may be easier to use. The following theorem shows that they are interchangeable, so either one can be used as the defmition oflim {(x) = l. X--->Xo 180 LIMITS AND CONTINUITY 181 Theorem 1 For any given f.
    [Show full text]
  • 1 Approximating Integrals Using Taylor Polynomials 1 1.1 Definitions
    Seunghee Ye Ma 8: Week 7 Nov 10 Week 7 Summary This week, we will learn how we can approximate integrals using Taylor series and numerical methods. Topics Page 1 Approximating Integrals using Taylor Polynomials 1 1.1 Definitions . .1 1.2 Examples . .2 1.3 Approximating Integrals . .3 2 Numerical Integration 5 1 Approximating Integrals using Taylor Polynomials 1.1 Definitions When we first defined the derivative, recall that it was supposed to be the \instantaneous rate of change" of a function f(x) at a given point c. In other words, f 0 gives us a linear approximation of f(x) near c: for small values of " 2 R, we have f(c + ") ≈ f(c) + "f 0(c) But if f(x) has higher order derivatives, why stop with a linear approximation? Taylor series take this idea of linear approximation and extends it to higher order derivatives, giving us a better approximation of f(x) near c. Definition (Taylor Polynomial and Taylor Series) Let f(x) be a Cn function i.e. f is n-times continuously differentiable. Then, the n-th order Taylor polynomial of f(x) about c is: n X f (k)(c) T (f)(x) = (x − c)k n k! k=0 The n-th order remainder of f(x) is: Rn(f)(x) = f(x) − Tn(f)(x) If f(x) is C1, then the Taylor series of f(x) about c is: 1 X f (k)(c) T (f)(x) = (x − c)k 1 k! k=0 Note that the first order Taylor polynomial of f(x) is precisely the linear approximation we wrote down in the beginning.
    [Show full text]
  • Series: Convergence and Divergence Comparison Tests
    Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions.
    [Show full text]
  • A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- Sity, Pomona, CA 91768
    A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule for differentiation. I showed my students the standard derivation of the Integration by Parts formula as presented in [1]: By the Product Rule, if f (x) and g(x) are differentiable functions, then d f (x)g(x) = f (x)g(x) + g(x) f (x). dx Integrating on both sides of this equation, f (x)g(x) + g(x) f (x) dx = f (x)g(x), which may be rearranged to obtain f (x)g(x) dx = f (x)g(x) − g(x) f (x) dx. Letting U = f (x) and V = g(x) and observing that dU = f (x) dx and dV = g(x) dx, we obtain the familiar Integration by Parts formula UdV= UV − VdU. (1) My student Victor asked if we could do a similar thing with the Quotient Rule. While the other students thought this was a crazy idea, I was intrigued. Below, I derive a Quotient Rule Integration by Parts formula, apply the resulting integration formula to an example, and discuss reasons why this formula does not appear in calculus texts. By the Quotient Rule, if f (x) and g(x) are differentiable functions, then ( ) ( ) ( ) − ( ) ( ) d f x = g x f x f x g x . dx g(x) [g(x)]2 Integrating both sides of this equation, we get f (x) g(x) f (x) − f (x)g(x) = dx.
    [Show full text]
  • Two Fundamental Theorems About the Definite Integral
    Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b].
    [Show full text]
  • Divergent and Conditionally Convergent Series Whose Product Is Absolutely Convergent
    DIVERGENTAND CONDITIONALLY CONVERGENT SERIES WHOSEPRODUCT IS ABSOLUTELYCONVERGENT* BY FLORIAN CAJOKI § 1. Introduction. It has been shown by Abel that, if the product : 71—« I(¥» + V«-i + ■•• + M„«o)> 71=0 of two conditionally convergent series : 71—0 71— 0 is convergent, it converges to the product of their sums. Tests of the conver- gence of the product of conditionally convergent series have been worked out by A. Pringsheim,| A. Voss,J and myself.§ There exist certain conditionally- convergent series which yield a convergent result when they are raised to a cer- tain positive integral power, but which yield a divergent result when they are raised to a higher power. Thus, 71 = «3 Z(-l)"+13, 7>=i n where r = 7/9, is a conditionally convergent series whose fourth power is con- vergent, but whose fifth power is divergent. || These instances of conditionally * Presented to the Society April 28, 1900. Received for publication April 28, 1900. fMathematische Annalen, vol. 21 (1883), p. 327 ; vol. 2(5 (1886), p. 157. %Mathematische Annalen, vol. 24 (1884), p. 42. § American Journal of Mathematics, vol. 15 (1893), p. 339 ; vol. 18 (1896), p. 195 ; Bulletin of the American Mathematical Society, (2) vol. 1 (1895), p. 180. || This may be a convenient place to point out a slight and obvious extension of the results which I have published in the American Journal of Mathematics, vol. 18, p. 201. It was proved there that the conditionally convergent series : V(_l)M-lI (0<r5|>). Sí n when raised by Cauchy's multiplication rule to a positive integral power g , is convergent whenever (î — l)/ï <C r ! out the power of the series is divergent, if (q— 1 )¡q > r.
    [Show full text]
  • Dirichlet’S Test
    4. CONDITIONALLY CONVERGENT SERIES 23 4. Conditionally convergent series Here basic tests capable of detecting conditional convergence are studied. 4.1. Dirichlet’s test. Theorem 4.1. (Dirichlet’s test) n Let a complex sequence {An}, An = k=1 ak, be bounded, and the real sequence {bn} is decreasing monotonically, b1 ≥ b2 ≥ b3 ≥··· , so that P limn→∞ bn = 0. Then the series anbn converges. A proof is based on the identity:P m m m m−1 anbn = (An − An−1)bn = Anbn − Anbn+1 n=k n=k n=k n=k−1 X X X X m−1 = An(bn − bn+1)+ Akbk − Am−1bm n=k X Since {An} is bounded, there is a number M such that |An|≤ M for all n. Therefore it follows from the above identity and non-negativity and monotonicity of bn that m m−1 anbn ≤ An(bn − bn+1) + |Akbk| + |Am−1bm| n=k n=k X X m−1 ≤ M (bn − bn+1)+ bk + bm n=k ! X ≤ 2Mbk By the hypothesis, bk → 0 as k → ∞. Therefore the right side of the above inequality can be made arbitrary small for all sufficiently large k and m ≥ k. By the Cauchy criterion the series in question converges. This completes the proof. Example. By the root test, the series ∞ zn n n=1 X converges absolutely in the disk |z| < 1 in the complex plane. Let us investigate the convergence on the boundary of the disk. Put z = eiθ 24 1. THE THEORY OF CONVERGENCE ikθ so that ak = e and bn = 1/n → 0 monotonically as n →∞.
    [Show full text]