Answered: Find a power series representation for…

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 32WE

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Question

Find a power series representation for the function of convergence.

f(x) = 1/1+4x^2 and determine the radius.

Expert Solution

Step 1

Given :

The function , $f (x) = \frac{1}{1 + 4 x^{2}}$

To find :

A power series representation for the function of convergence and the radius of convergence of the function , $f (x) = \frac{1}{1 + 4 x^{2}}$

Step 2

$f (x) = \frac{1}{1 + 4 x^{2}}$

The function $f (x) = \frac{1}{1 + 4 x^{2}}$ can be written as

$f (x) = \frac{1}{1 + 4 x^{2}} = \frac{1}{1 - (- 4 x^{2})}$

Comparing this function with the geometric series, $\frac{a}{1 - r}$ , where a is the first term and r is the common ratio.

Here, a = 1, and r = $- 4 x^{2}$

Therefore, f (x) can be written as a geometric series ,

$\begin{array}{rcl} f (x) & = & \sum_{n = 0}^{\infty} a r^{n} \\ = & \sum_{n = 0}^{\infty} (1) (- 4 x^{2})^{n} \\ = & \sum_{n = 0}^{\infty} {(- 1)}^{n} 4^{n} (x^{2})^{n} \\ = & \sum_{n = 0}^{\infty} {(- 1)}^{n} 4^{n} x^{2 n} \end{array}$

Therefore, the power series representation of $f (x) = \begin{array}{rcl} \end{array} \begin{array}{rcl} \sum_{n = 0}^{\infty} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} (- 1) \end{array} \begin{array}{rcl} \end{array}$ ⁿ 4ⁿ x²ⁿ.

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