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In this exercise you will find the sum of the series S(-1)3n+3 22n+3 (n + 1) n=0 1 - for -1 < x < 1. Show that (-1, 1) is the interval of convergence. 1+ x2 (a) Find a power series centered at 0 which converges to (b) Integrate the power series from part (a), and find a power series centered at 0 that converges to x In(1 + x²) on some interval. Find the interval of convergence of this power series. (-1)3n+3 22n+3 (n + 1) (c) Explain why > is convergent, and find its sum. n=0

In this exercise you will find the sum of the series S(-1)3n+3 22n+3 (n + 1) n=0 1 - for -1 < x < 1. Show that (-1, 1) is the interval of convergence. 1+ x2 (a) Find a power series centered at 0 which converges to (b) Integrate the power series from part (a), and find a power series centered at 0 that converges to x In(1 + x²) on some interval. Find the interval of convergence of this power series. (-1)3n+3 22n+3 (n + 1) (c) Explain why > is convergent, and find its sum. n=0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Power Series

In this exercise you will find the sum of the series (-1)3n+3 22n+3 (п + 1) 00 n=0 1 for -1 < x < 1. Show that (-1, 1) is the interval of convergence. (a) Find a power series centered at 0 which converges to 1+ x2 (b) Integrate the power series from part (a), and find a power series centered at 0 that converges to x In(1 + x²) on some interval. Find the interval of convergence of this power series. Š (-1)3n+3 22n+3 (n + 1) (c) Explain why is convergent, and find its sum. n=0
Transcribed Image Text:In this exercise you will find the sum of the series (-1)3n+3 22n+3 (п + 1) 00 n=0 1 for -1 < x < 1. Show that (-1, 1) is the interval of convergence. (a) Find a power series centered at 0 which converges to 1+ x2 (b) Integrate the power series from part (a), and find a power series centered at 0 that converges to x In(1 + x²) on some interval. Find the interval of convergence of this power series. Š (-1)3n+3 22n+3 (n + 1) (c) Explain why is convergent, and find its sum. n=0
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