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Q. 20

Expert-verified

Calculus

Found in: Page 692

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k} {(\frac{1}{π})}^{k}$

The required answer is $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k} {(\frac{1}{π})}^{k} = - \ln (1 + \frac{1}{π})$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given series is $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k} {(\frac{1}{π})}^{k}$

Step 2. Explanation

The series can be rewritten as $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k} {(\frac{1}{π})}^{k} = - \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1}}{k} {(\frac{1}{π})}^{k}$

The Maclaurin series for the function role="math" localid="1649321001787" $f (x) = \ln (1 + x) is \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1}}{k} {(x)}^{k}$

So, the series $- \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1}}{k} {(\frac{1}{π})}^{k}$ is the maclaurin series for $- \ln (1 + x) a t x = \frac{1}{π}$

Since, $- \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1}}{k} {(\frac{1}{π})}^{k} = - \ln (1 + \frac{1}{π})$

Thus,

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k} {(\frac{1}{π})}^{k} = - \ln (1 + \frac{1}{π})$

Most popular questions for Math Textbooks

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:

$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Graph the first four terms in the sequence of partial sums of $J_{1} (x)$ .

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22. $\sum_{k = 0}^{\infty} \frac{{(- π)}^{k}}{k!}$

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $\pm p$ , then the series converges absolutely at the other value as well.

Let $a_{k} \neq 0$ for each value of $k$ , and let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a positive and finite radius of convergence $p$ . What is the radius of convergence of the power series $\sum_{k = 0}^{\infty} \frac{1}{a_{k}} x^{k}$ ?

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k + 1}}{(2 k + 1)!}$

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