Answered: Find The Powet series for

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 49E

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Expert Solution

Step 1

Given:

$Here f (x) = \frac{x}{{(1 - x)}^{2}}$

To find:

we have to find the power series for f(x)

Step 2

$Here f (x) = \frac{x}{{(1 - x)}^{2}} we know the power series for \frac{1}{1 - x} is \sum_{n = 0}^{N} x^{n}$

$L e t \sum_{n = 0}^{N} x^{n} = \frac{1}{1 - x}$

Differentiate on both side w.r.t.x,we get

$\frac{d}{d x} [\sum_{n = 0}^{N} x^{n}] = \frac{d}{d x} [\frac{1}{1 - x}] \sum_{n = 1}^{N} n x^{n - 1} = \frac{1}{{(1 - x)}^{2}}$

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