**Instruction on Power Series**To solve the problem, you are provided with the power series:\[\frac{1}{1 + x} = \sum_{n=0}^{\infty} (-1)^n x^n, \quad |x| < 1\]Use this series to find a power series for the function $ h(x) $, centered at 0:\[h(x) = \frac{-2}{x^2 - 1} = \frac{1}{1 + x} + \frac{1}{1 - x}\]*Goal:* Express $ h(x) $ in a power series form:\[h(x) = \sum_{n=0}^{\infty} \]\[ \underline{\hspace{5 cm}} \]**Task:** Determine the interval of convergence. (Provide your answer using interval notation.)\[ \underline{\hspace{5 cm}} \]

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Use the power series 00 1 + x E(-1)^x?, \xl < 1 n = 0 to find a power series for the function, centered at 0. -2 1 1. h(x) x2 - 1 1 + X 1- X h(x) n- 0 Determine the interval of convergence. (Enter your answer using interval notation.)

Use the power series 00 1 + x E(-1)^x?, \xl < 1 n = 0 to find a power series for the function, centered at 0. -2 1 1. h(x) x2 - 1 1 + X 1- X h(x) n- 0 Determine the interval of convergence. (Enter your answer using interval notation.)

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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Question

**Instruction on Power Series**

To solve the problem, you are provided with the power series:

\[
\frac{1}{1 + x} = \sum_{n=0}^{\infty} (-1)^n x^n, \quad |x| < 1
\]

Use this series to find a power series for the function $ h(x) $, centered at 0:

\[
h(x) = \frac{-2}{x^2 - 1} = \frac{1}{1 + x} + \frac{1}{1 - x}
\]

*Goal:* Express $ h(x) $ in a power series form:

\[
h(x) = \sum_{n=0}^{\infty} \]
\[ \underline{\hspace{5 cm}} \]

**Task:** Determine the interval of convergence. (Provide your answer using interval notation.)

\[ \underline{\hspace{5 cm}} \]

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