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Find power series with the following properties. Justify your answers completely. (a) A power series that converges at r =2 and x = 4, but not at x = 1.

Find power series with the following properties. Justify your answers completely. (a) A power series that converges at r =2 and x = 4, but not at x = 1.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find power series with the following properties. Justify your answers completely. (a) A power series that converges at r = 2 and x = 4, but not at x = 1.
Transcribed Image Text:Find power series with the following properties. Justify your answers completely. (a) A power series that converges at r = 2 and x = 4, but not at x = 1.
Expert Solution
Step 1

Use and manipulate geometric series \frac{1}{1-z}=\sum_{n=0}^{\infty}z^n,~\left|{z}\right|<1 for the first three of these.

a) Consider the functionf(z)=\frac{1}{z-1}+\frac{1}{5-z}

Expand this function in series form in the annulus 1<|z|<5 centered at z=0, by considering each term separately.

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