Question

Does (-2)^n/( 3^n+1) converge or diverge?

Answer 1

$\left|\dfrac{(-2)^n}{3^n+1}\right|=\dfrac{2^n}{3^n+1}$

As $n\to\infty$, the sequence $a_n=\left(\dfrac23\right)^n$ converges to zero.

If you're talking about the infinite series

$\displaystyle\sum_{n\ge0}\dfrac{(-2)^n}{3^n+1}$

well we've shown by comparison that this series must also converge because we know any geometric series $\sum\limits_n r^n$ will converge as long as $|r|$.