Answer
The series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$ converges.
Work Step by Step
In the series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$, we have the geometric series $\sum_{n=1}^{\infty} (\frac{3 }{5})^n$, which converges since $r=3/5\lt 1$. For the series $\sum_{n=1}^{\infty} \frac{(-2)^n}{5^{n}}$, the positive term $b_n=(\frac{2}{5})^n$; then we have
$$\lim_{n\to \infty}b_n=\lim_{n\to \infty}(\frac{2}{5})^n=0$$
since $2/5\lt 1$. Using the alternating series test, the series $\sum_{n=1}^{\infty} \frac{(-2)^n}{5^{n}}$ converges. Hence the series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$ converges.
