SENGAMALA THAYAAR EDUCATIONAL TRUST WOMEN’S COLLEGE (Accredited with ‘A ’grade by NAAC & An ISO 9001: 2015 Certified Institution) SUNDARAKKOTTAI, MANNARGUDI-614 016 Sequences and Series – Unit – I Study Material for II B.Sc., Mathematics Students Prepared by Dr.S.Suganthi, M.Sc., M.Phil., Ph.d, SET. Assistant Professor of Mathematics STET Women’s College CORE COURSE V SEQUENCES AND SERIES 16SCCMM5 OBJECTIVES : 1. To lay a good foundation for classical analysis 2. To study the behavior of sequences and series. Unit I Sequences – Bounded Sequences – Monotonic Sequences – Convergent Sequence – Divergent Sequences – Oscillating sequences Unit II Algebra of Limits – Behavior of Monotonic functions Unit III Some theorems on limits – sub sequences – limit points : Cauchy sequences Unit IV Series – infinite series – Cauchy’s general principal of convergence – Comparison – test theorem and test of convergence using comparison test (comparison test statement only, no proof) Unit V Test of convergence using D Alembert’s ratio test – Cauchy’s root test – Alternating Series – Absolute Convergence (Statement only for all tests) Book for Study Dr. S.Arumugam & Mr.A.Thangapandi Isaac Sequences and Series – New Gamma Publishing House – 2002 Edition. Unit I : Chapter 3 : Sec. 3.0 – 3.5 Page No : 39-55 Unit II : Chapter 3 : Sec. 3.6, 3.7 Page No:56 – 82 Unit III : Chapter 3 : Sec. 3.8-3.11, Page No:82-102 Unit IV : Chapter 4 : Sec. (4.1 & 4.2) Page No : 112-128. Unit V : Relevant part of Chapter 4 and Chapter 5: Sec. 5.1 & 5.2 Page No:157-167. Book for Reference 1. Algebra – Prof. S.Surya Narayan Iyer 2. Algebra – Prof. M.I.Francis Raj Sequences and Series UNIT - I Set Definition: A set is a collection of well - defined and distinct objects. Example The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on Note Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Function Definition: A function is more formally defined given a set of inputs X (domain) and a set of possible outputs Y (codomain) as a set of ordered pairs (x,y) where x∈X and y∈Y, subject to the restriction that there can be only one ordered pair with the same value of x. We can write the statement that f is a function from X to Y using the function notation f : X →Y. 3.1 Sequences Definition: Let f : N →R be a function and let f(n) = an . Then a1, a2, a3, a4,……… an,…….is called the Sequence in R determined by the function f and is denoted by (an). an is called the nth term of the sequence. Note : The range of the function f, which is a subset of R, is called the range of the sequence. Examples 1. The function f : N →R given by f(n) = n determines the sequence 1,2,3,……..n,… (ie) f(1) = 1, f(2) = 2, f(3) = 3,….. f(4) = 4,….. 2. The function f: N →R given by f(n) = n2 determines the sequence 1,4,9,….n2,… (ie) f(1) = 1, f(2) = 4, f(3) = 9,….. f(4) = 16,….. 3. The function f: N →R given by f(n) = (-1)n determines the sequence −1,1, −1,1……..…Thus the terms of a sequence need not be distinct. The range of this sequence is {ퟏ, −ퟏ}. 4. The Sequence ( (-1)n+1 ) is given by 1, -1, 1, -1…..The range of this sequence is also {ퟏ, −ퟏ}. Note The sequence ( (-1)n ) and ( (-1)n+1 ) are different . The first sequence starts with -1 and the second sequence starts with 1. 5. The Constant function f: N →R given by f(n) = 1 determines the sequence 1,1,1,1… such a sequence is called a constant sequence 1 푛 푖푓 푛푖푠 푒푣푒푛 2 6. The function f: N →R given by f(n) = { 1 determines the (1 − 푛) 푖푓 푛푖푠 표푑푑 2 sequence 0, 1, -1, 2, -2,…….,n, -n,……. (ie) f(1) = 0, f(2) = 1, f(3) = -1, f(4) = 2, f(5) = -2,………… The range of this sequence is Z (integers). 푛 1 2 푛 7. The function f: N →R given by f(n) = determines the sequence , , …., ,… 푛+1 2 3 푛+1 8. The function f: N →R given by f(n) =2n+3 determines the sequence 5, 7, 9, 11, … 1 1 1 1 9. The function f: N →R given by f(n) = determines the sequence1, , , …., ,… 푛 2 3 푛 10. Let xєR. The function f: N →R given by f(n) = xn-1 determines the geometric sequence 1, x, x2, ………xn,….. 11. The sequence (-n) is given by -1, -2, -3,….-n,…The range of this sequence is the set of all negative integers. Exercises 1.Write the first five terms of each of the following sequences. (−1)푛 2 1 푐표푠푛푥 (−1)푛+1 (a) ( ) (b) ( (1 − )) (c) ( ) (d) ( ) 푛 3 10푛 푛2+푥2 푛! 1−(−1)푛 2푛2+1 (e) ( ) (f) ( ) (g) (n!) 푛3 2푛2−1 푛 푖푓 푛 푖푠 표푑푑 (h) f(n) = {1 푖푓 푛 푖푠 푒푣푒푛 푛 2. Determine the range of the following sequences. (a) (n) (b) (2n) (c) (2n-1) (d) (1+(-1)n) (e) The constant sequence a, a, a,……. 3.2 Bounded Sequences Definition: A sequence (an) is said to be Bounded Above if there exists a real number k such that an≤k for every n∈N. k is called an upper bound of the sequence (an). A sequence (an) is said to be Bounded Below if there exists a real number k such that an≥k for every n∈N. k is called an lowerr bound of the sequence (an). A sequence (an) is said to be Bounded if it is both bounded above and bounded below. Note 1. A sequence (an) is bounded iff there exists a real number k≥ 0 Such that ⃒an⃒≤ k for all n. (ie) -k ≤ an ≤ k Least upper bound: An upper bound for a set A which belongs to A is a greatest element of A (or a maximum element). Notice that A can have only one greatest element. The least upper bound of A is also called the supremum of A. It can be written sup(A) or lub(A). Greatest lower bound: An lower bound for a set A which belongs to A is a least element of A (or a minimum element). Notice that A can have only one least element. The greatest lower bound of A is also called the infimum of A. It can be written inf(A) or glb(A). `Examples 1 1 1 1. Consider the sequences 1, , , …….., ,………..Here 1 is l.u.b and 0 is the g .l.b. 2 3 푛 It is bounded sequence. 2. The sequence 1,2,3,….,n,…. is bounded below but not bounded above. 1 is the g.l.b of the sequence. 3. The sequence -1,-2,-3,….,-n,…. is bounded above but not bounded below. -1 is the l.u.b of the sequence. 4. 1,-1,1,-1,…..is a bounded sequences. Here 1 is the l.u.b., -1 is the g.l.b. of the sequence. 5. Any constant sequence is a bounded sequence. (eg) The sequence 2,2,2,2,….here l.u.b and g.l.b is 2 (ie) l.u.b = g.l.b = 2 Exercises 1. Give examples of sequences (an) such that (a). (an) is bounded below but not bounded above. (b). (an) is bounded above but not bounded below. (c). (an) is a bounded sequence. (d). (an) is neither bounded above nor bounded below. 2. Determine the l.u.b and g.l.b of the following sequences if they exist (a). 2, -2, 1, -1, 1, -1, ……………. 1 1 1 1 (b). 1, , , , …….., ,…… √2 √3 √4 √푛 1 2 3 푛 (c). , , , ….., ,…………. 2 3 4 푛+1 (d). 1, -1, 2, -2, 3,-3, ……….n, -n,……… 1 1 1 1 (e) 1, , 3, , 5, , ………………(2n-1), ,……….. 2 4 6 2푛 1 1 1 (f). 1, ,1, , 1, ,…………… 2 3 4 (g). (1+n+n2) (h). (-n2) 3.3 Monotonic Sequences Definitions A sequence (an) is said to be monotonic increasing if an≤ an+1 for all n. A sequence (an) is said to be monotonic decreasing if an≥ an+1 for all n. A sequence (an) is said to be strictly monotonic increasing if an< an+1 for all n . A sequence (an) is said to be strictly monotonic decreasing if an> an+1 for all n. A sequence (an) is said to be monotonic if it is either monotonic increasing or monotonic decreasing. Examples 1. 1, 2,2, 3, 3, 3, 4, 4, 4, 4,,……….is a monotonic increasing sequence.(ie) an≤ an+1 2. 1,2,3,4,………,n, …….is a strictly monotonic increasing sequence. (ie) an< an+1 1 1 1 3. 1, , , …….., ,………..is a strictly monotonic decreasing sequence. (ie) an> an+1 2 3 푛 4. The sequence (an) given by 1, -1, 1, -1, 1,….is neither monotonic increasing sequence nor monotonic decreasing sequence. Hence (an) is not a monotonic sequence. ퟐ풏−ퟕ 5. Prove ( ) is a monotonic increasing sequence. ퟑ풏+ퟐ Proof: 2푛−7 2(푛+1)−7 Ler an = , an+1 = 3푛+2 3(푛+1)+2 2푛−7 2(푛+1)−7 Now an− an+1 = - 3푛+2 3(푛+1)+2 (2푛−7)(3푛+5)−(2푛−5)(3푛+2) = (3푛+2)(3푛+5) −25 = < 0.