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