Power Series and Taylor’s Series

Dr. Kamlesh Jangid

Department of HEAS (Mathematics)

Rajasthan Technical University, Kota-324010, India

E-mail: [email protected]

Dr. Kamlesh Jangid (RTU Kota) Series 1 / 19 Outline

Outline

1 Introduction

2 Series

3 Convergence and Divergence of Series

4 Power Series

5 Taylor’s Series

Dr. Kamlesh Jangid (RTU Kota) Series 2 / 19 Introduction

Overview

Everyone knows how to add two numbers together, or even several.

But how do you add infinitely many numbers together ?

In this lecture we answer this question, which is part of the theory of infinite sequences and series.

An important application of this theory is a method for representing a known differentiable function f (x) as an infinite sum of powers of x, so it looks like a "polynomial with infinitely many terms."

Dr. Kamlesh Jangid (RTU Kota) Series 3 / 19 Series

Definition A sequence of real numbers is a function from the set N of natural numbers to the set R of real numbers.

If f : N → R is a sequence, and if an = f (n) for n ∈ N, then we write the sequence f as {an}. A sequence of real numbers is also called a real sequence.

Example

(i) {an} with an = 1 for all n ∈ N-a constant sequence.

n−1 1 2 (ii) {an} = { n } = {0, 2 , 3 , ···},

n+1 1 1 1 (iii) {an} = {(−1) n } = {1, − 2 , 3 , ···}.

Dr. Kamlesh Jangid (RTU Kota) Series 4 / 19 Series

Definition A series of real numbers is an expression of the form

a1 + a2 + a3 + ··· ,

∞ X or more compactly as an, where {an} is a sequence of real n=1 numbers.

The number an is called the n-th term of the series and the n X sequence sn := ai is called the n-th partial sum of the series i=1 ∞ X an. n=1

Dr. Kamlesh Jangid (RTU Kota) Series 5 / 19 Convergence and Divergence of Series

Convergence and divergence

Definition ∞ X A series an is said to converge to a real number s, if the n=1 sequence {sn} of partial sums of the series converges (to s ∈ R). ∞ ∞ X X If an converges to s, then we write an = s. n=1 n=1 A series which does not converge is called a divergent series.

Dr. Kamlesh Jangid (RTU Kota) Series 6 / 19 Convergence and Divergence of Series Theorem (A necessary condition) ∞ X If series an converges, then an → 0 as n → ∞. Converse does n=1 not hold. P∞ 1 Ex.: n=1 n diverges, whereas an → 0.

Corollary

Suppose {an} is a sequence of positive terms such that an+1 > an ∞ X for all n ∈ N. Then the series an diverges. n=1

Example P∞ n The above theorem and corollary shows, that the series n=1 n+1 diverges.

Dr. Kamlesh Jangid (RTU Kota) Series 7 / 19 Convergence and Divergence of Series Example P∞ n−1 Consider the geometric series n=1 a q , where a, q ∈ R. n−1 Note that sn = a + aq + ··· + aq for n ∈ N. Clearly, if a = 0, then sn = 0 for all n ∈ N. Hence, assume that a 6= 0. Then we have   n a, if q = 1, sn = a(1−qn)  1−q , if q 6= 1.

Thus, if q = 1, then {sn} is not bounded; hence not convergent. If q = −1, then we have   a, if n odd, sn =  0, if n even.

Dr. Kamlesh Jangid (RTU Kota) Series 8 / 19 Convergence and Divergence of Series

Thus, {sn} diverges for q = −1 as well. Now, assume that |q|= 6 1. In this case, we have

a |a| n sn − = |q| . 1 − q |1 − q|

a This shows that, if |q| < 1, then {sn} converges to 1−q , and if

|q| > 1, then {sn} is not bounded, hence diverges.

Dr. Kamlesh Jangid (RTU Kota) Series 9 / 19 Convergence and Divergence of Series

Some tests for convergence Theorem (Comparison test)

Suppose {an} and {bn} are sequences of non-negative terms, and an ≤ bn for all n ∈ N. Then, P∞ P∞ (i) n=1 bn converges ⇒ n=1 an converges, P∞ P∞ (ii) n=1 an diverges ⇒ n=1 bn diverges.

Example ∞ √1 1 P 1 Since n ≥ n for all n ∈ N, and since the series n=1 n diverges, ∞ P √1 it follows from the above theorem that the series n=1 n also diverges.

Dr. Kamlesh Jangid (RTU Kota) Series 10 / 19 Convergence and Divergence of Series

Theorem (Limit form test)

Suppose {an} and {bn} are sequences of non-negative terms. (a) Suppose ` := lim an exists. Then we have the following: n→∞ bn P∞ P∞ (i) If ` > 0, then n=1 bn converges ⇔ n=1 an converges. P∞ P∞ (ii) If ` = 0, then n=1 bn converges ⇒ n=1 an converges. (b) Suppose lim an = ∞ exists. n→∞ bn P∞ P∞ Then n=1 an converges ⇒ n=1 bn converges.

Dr. Kamlesh Jangid (RTU Kota) Series 11 / 19 Convergence and Divergence of Series

Theorem (De’Alembert’s ratio test)

Suppose {an} is a sequences of non-negative terms such that lim an+1 = ` exists. Then we have the following: n→∞ an P∞ (i) If ` < 1, then the series n=1 an converges. P∞ (ii) If ` > 1, then the series n=1 an diverges.

Example P∞ xn For every x ∈ R, the series n=1 n! converges: n Here, a = x . Hence an+1 = x ∀n ∈ . Hence, it follows that n n! an n+1 N lim an+1 = 0, so that by De’Alemberts test, the series n→∞ an converges.

Dr. Kamlesh Jangid (RTU Kota) Series 12 / 19 Convergence and Divergence of Series

Example (Harmonic series (P-series)) X 1 The harmonic series converges if p > 1, and diverges if np 0 < p ≤ 1.

Dr. Kamlesh Jangid (RTU Kota) Series 13 / 19 Power Series

Power Series: A series of the form

∞ n X n a0 + a1x + ··· + anx + ··· = anx n=0 is called a power series in x. Radius of convergence and Interval of convergence: A constant R > 0 is said to be the radius of convergence of the P∞ n power series n=0 anx , if the power series converges for |x| < R and diverges for |x| > R. The series may or may not converge for |x| = R. The interval |x| < R or −R < x < R, for which the power series P∞ n n=0 anx is convergent, is called the interval of convergence. Dr. Kamlesh Jangid (RTU Kota) Series 14 / 19 Power Series

Remark P∞ n (i) If R = 0, the series n=0 anx converges for x = 0. Thus, every power series is convergent for x = 0.

P∞ n (ii) If R = ∞, the series n=0 anx converges for all real, i.e., everywhere convergent.

(iii) Radius of convergence R is calculated as

an lim n→∞ an+1

i.e.,

an R = lim n→∞ an+1

Dr. Kamlesh Jangid (RTU Kota) Series 15 / 19 Power Series

Example P∞ n x2n+1 Determine the interval of convergence of n=0(−1) 2n+1 .

Solution (−1)n Here an = 2n+1 , then

an 2n+3 R = limn→∞ | | = limn→∞ | | = 1. an+1 2n+1 Thus, the interval of convergence is |x| < 1 or −1 < x < 1.

Dr. Kamlesh Jangid (RTU Kota) Series 16 / 19 Taylor’s Series

Taylor’s series: Let f (x) be a real valued function that is infinitely differentiable at x = a, then

f (n)(a) f (x) = f (a) + (x − a)f 0(a) + ··· + (x − a)n + ··· n! ∞ X f (n)(a) = (x − a)n. n! n=0

Dr. Kamlesh Jangid (RTU Kota) Series 17 / 19 Taylor’s Series

Convergence of Exponential series: x P∞ xn We have e = n=0 n! .

1 an (n+1)! Here an = , then R = limn→∞ | | = limn→∞ | | = ∞. n! an+1 n! Hence the series is convergent everywhere.

Convergence of Logarithmic series: P∞ n−1 xn We have log(1 + x) = n=1(−1) n . n−1 (−1) an (n+1) Here an = , then R = limn→∞ | | = limn→∞ | | = 1. n an+1 n Hence the series is convergent for |x| < 1 and diverges for |x| > 1.

Convergence of Trigonometric series: Try your self

Dr. Kamlesh Jangid (RTU Kota) Series 18 / 19 Taylor’s Series

For the video lecture use the following link https://youtube.com/channel/ UCk9ICMqdkO0GREITx-2UaEw

THANK YOU

Dr. Kamlesh Jangid (RTU Kota) Series 19 / 19