Answer
Converges
Work Step by Step
Apply the Ratio test to the series
$\Sigma^{\infty}_{n=1} n(\frac{7}{8})^n$
$\lim\limits_{n \to \infty} |\frac{(n+1)(\frac{7}{8})^{n+1}}{n(\frac{7}{8})^n}|$, Absolute value not needed because all values are positive
$ \lim\limits_{n \to \infty} \frac{7(n+1)}{8n} = \frac{7}{8} <1$
Because the answer is less than 1, the series converges
