## Calculus 8th Edition

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