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Power Series and Taylor's Series

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Power Series and Taylor's Series 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 2 N, then we write the sequence f as fang. A sequence of real numbers is also called a real sequence. Example (i) fang with an = 1 for all n 2 N-a constant sequence. n−1 1 2 (ii) fang = f n g = f0; 2 ; 3 ; · · · g; n+1 1 1 1 (iii) fang = f(−1) n g = f1; − 2 ; 3 ; · · · g. Dr. Kamlesh Jangid (RTU Kota) Series 4 / 19 Series Definition A series of real numbers is an expression of the form a1 + a2 + a3 + ··· ; 1 X or more compactly as an, where fang 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 1 X an. n=1 Dr. Kamlesh Jangid (RTU Kota) Series 5 / 19 Convergence and Divergence of Series Convergence and divergence Definition 1 X A series an is said to converge to a real number s, if the n=1 sequence fsng of partial sums of the series converges (to s 2 R). 1 1 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) 1 X If series an converges, then an ! 0 as n ! 1. Converse does n=1 not hold. P1 1 Ex.: n=1 n diverges, whereas an ! 0. Corollary Suppose fang is a sequence of positive terms such that an+1 > an 1 X for all n 2 N. Then the series an diverges. n=1 Example P1 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 P1 n−1 Consider the geometric series n=1 a q , where a; q 2 R. n−1 Note that sn = a + aq + ··· + aq for n 2 N. Clearly, if a = 0, then sn = 0 for all n 2 N. Hence, assume that a 6= 0. Then we have 8 < n a; if q = 1; sn = a(1−qn) : 1−q ; if q 6= 1: Thus, if q = 1, then fsng is not bounded; hence not convergent. If q = −1, then we have 8 < a; if n odd; sn = : 0; if n even: Dr. Kamlesh Jangid (RTU Kota) Series 8 / 19 Convergence and Divergence of Series Thus, fsng diverges for q = −1 as well. Now, assume that jqj 6= 1. In this case, we have a jaj n sn − = jqj : 1 − q j1 − qj a This shows that, if jqj < 1, then fsng converges to 1−q , and if jqj > 1, then fsng 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 fang and fbng are sequences of non-negative terms, and an ≤ bn for all n 2 N. Then, P1 P1 (i) n=1 bn converges ) n=1 an converges, P1 P1 (ii) n=1 an diverges ) n=1 bn diverges. Example 1 p1 1 P 1 Since n ≥ n for all n 2 N, and since the series n=1 n diverges, 1 P p1 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 fang and fbng are sequences of non-negative terms. (a) Suppose ` := lim an exists. Then we have the following: n!1 bn P1 P1 (i) If ` > 0, then n=1 bn converges , n=1 an converges. P1 P1 (ii) If ` = 0, then n=1 bn converges ) n=1 an converges. (b) Suppose lim an = 1 exists. n!1 bn P1 P1 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 fang is a sequences of non-negative terms such that lim an+1 = ` exists. Then we have the following: n!1 an P1 (i) If ` < 1, then the series n=1 an converges. P1 (ii) If ` > 1, then the series n=1 an diverges. Example P1 xn For every x 2 R, the series n=1 n! converges: n Here, a = x . Hence an+1 = x 8n 2 : Hence, it follows that n n! an n+1 N lim an+1 = 0, so that by De’Alemberts test, the series n!1 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 1 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 P1 n power series n=0 anx , if the power series converges for jxj < R and diverges for jxj > R. The series may or may not converge for jxj = R. The interval jxj < R or −R < x < R, for which the power series P1 n n=0 anx is convergent, is called the interval of convergence. Dr. Kamlesh Jangid (RTU Kota) Series 14 / 19 Power Series Remark P1 n (i) If R = 0, the series n=0 anx converges for x = 0. Thus, every power series is convergent for x = 0. P1 n (ii) If R = 1, the series n=0 anx converges for all real, i.e., everywhere convergent. (iii) Radius of convergence R is calculated as an lim n!1 an+1 i.e., an R = lim n!1 an+1 Dr. Kamlesh Jangid (RTU Kota) Series 15 / 19 Power Series Example P1 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!1 j j = limn!1 j j = 1. an+1 2n+1 Thus, the interval of convergence is jxj < 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! 1 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 P1 xn We have e = n=0 n! . 1 an (n+1)! Here an = , then R = limn!1 j j = limn!1 j j = 1. n! an+1 n! Hence the series is convergent everywhere. Convergence of Logarithmic series: P1 n−1 xn We have log(1 + x) = n=1(−1) n . n−1 (−1) an (n+1) Here an = , then R = limn!1 j j = limn!1 j j = 1. n an+1 n Hence the series is convergent for jxj < 1 and diverges for jxj > 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.
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