Answered: Σ-2 k=0

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.5: Other Types Of Equations

Problem 11E

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Question

Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)

Expert Solution

Step 1

Consider that $\sum_{k = 0}^{\infty} {(\sqrt{x} - 2)}^{k}$ is the power series expansion of $f (x) = \frac{1}{1 - x}$ .

Then, $f (x) = \frac{1}{1 - x}$ ...…(1)

Substitute x by $(\sqrt{x} - 2)$ as,

$\sum_{k = 0}^{\infty} {(\sqrt{x} - 2)}^{k} = \frac{1}{1 - (\sqrt{x} - 2)} = \frac{1}{3 - \sqrt{x}}$

Then, $\sum_{k = 0}^{\infty} {(\sqrt{x} - 2)}^{k}$ is the power series expansion of $\frac{1}{3 - \sqrt{x}}$ .

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