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oj vergence of the power series. 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = d Σ k=0 dz 8] k k! k=0 k -Σ = k! k=0 dzk dz k!' by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.

oj vergence of the power series. 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = d Σ k=0 dz 8] k k! k=0 k -Σ = k! k=0 dzk dz k!' by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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Let us recall from recitation (or Section V.3 of [G]) that if the sequence (cn)o of complex coefficients is such that the limit Cn R = lim n→∞ Cn+1 exists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series n=0 in the complex variable w converges absolutely for |w| < R and does not converge for |w| > R. We call R the radius of convergence of the power series. 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz GnWn -e² e² = (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = k=0 zk k! k d «Σ-Σ = dz k! k=0 ∞ dzk dz k!' k=0 by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.
Transcribed Image Text:Let us recall from recitation (or Section V.3 of [G]) that if the sequence (cn)o of complex coefficients is such that the limit Cn R = lim n→∞ Cn+1 exists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series n=0 in the complex variable w converges absolutely for |w| < R and does not converge for |w| > R. We call R the radius of convergence of the power series. 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz GnWn -e² e² = (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = k=0 zk k! k d «Σ-Σ = dz k! k=0 ∞ dzk dz k!' k=0 by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.
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