Question

Does the following infinite series converge or diverge?

Answer 1

The nth term is (presuming the denominator on the first is a typo)

$-\dfrac{n}{(-3)^n}$

The ratio of successive terms is

$\dfrac{ \frac{n+1}{(-3)^(n+1)}}{\frac{n}{(-3)^n}} = \frac{n+1}{-3n}$

The limit is -1/3, absolute value less than 1, so the series converges.  It has a sum -- that's what converges means.