Answer
Divergent
Work Step by Step
Given: $\Sigma_{n=1 }^{\infty}\frac{n\pi^{n}}{(-3)^{n-1}}$
Re-write the given series.
$\Sigma_{n=1 }^{\infty}\frac{n\pi^{n}}{(-3)^{n-1}}=\Sigma_{n=1 }^{\infty}(-1)^{n}3n(\pi/3)^{n}$
$L=\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}\frac{(-1)^{n+1}3(n+1)(\pi/3)^{n+1} }{(-1)^{n}3n(\pi/3)^{n}}$
$=\lim\limits_{n \to \infty}\frac{n+1}{n}\frac{\pi}{3}$
$=\lim\limits_{n \to \infty}[1+\frac{1}{n}]\frac{\pi}{3}$
$=[1+0]\frac{\pi}{3}$
$=\frac{\pi}{3}\gt 1$
Hence, the series diverges by ratio test.