Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.2 Series - 11.2 Exercises - Page 756: 57


series converges, and the sum is $\frac{-5x}{1+5x}$

Work Step by Step

Given: $\Sigma^{\infty}_{n=1} (-5)^{n}x^{n}$ $\Sigma^{\infty}_{n=1} (-5)^{n}x^{n} = \Sigma^{\infty}_{n=1}(-5x)^{n}$ Here, $a=-5x$ and $r=-5x$ $|r| \lt 1$ $|-5x| \lt 1$ $|5x| \lt 1$ $-\frac{1}{5} \lt x \lt \frac{1}{5}$ Therefore the series converges, and the sum is $$\Sigma^{\infty}_{n=1} (-5x)^{n} = \frac{a}{1-r}$$ $$=\frac{-5x}{1+5x}$$
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