Answered: Use the Geometric Series Test to help…

Use the Geometric Series Test to help you find a power series representation of f(x) = x/(2 + x^3) centered at 0. Find the interval and radius of convergence

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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Question

Use the Geometric Series Test to help you find a power series representation of f(x) = x/(2 + x^3)
centered at 0. Find the interval and radius of convergence.

Expert Solution

Given a function

$f (x) = \frac{x}{{(2 + x)}^{3}}$

To Find: power of series

Note:

$The geometric power series is a series of the form \frac{a}{1 - r} = \sum_{n = 0}^{\infty} {ar}^{n}, |r| < 1 Where a is the first term and r is the common ratio .$

$This implies f (x) = \frac{x}{2 + x^{3}} = \frac{x}{2} \cdot (\frac{1}{1 + \frac{x^{3}}{2}})$

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