Calculus II Series Help: $\sum _{n=1}^{\infty} \left((-2)^{n-2}+\frac{1}{5^{2n}}\right)/3^{n+1}$

Question

I am trying to solve this problem and I am having a lot of difficulty with it. The question is as follows:

Determine if the series converges or diverges. If convergent, find the exact sum.

$$\sum _{n=1}^{\infty}\:\left(\frac{\left(-2\right)^{n-2}+\frac{1}{5^{2n}}}{3^{n+1}}\right)$$

Any help would be much appreciated. Thank you.

Answer 1

Hint: $$\sum _{n=1}^{\infty }\:\left(\frac{\left(-2\right)^{n-2}+\frac{1}{5^{2n}}}{3^{n+1}}\right)=\frac{1}{12}\sum _{n=1}^{\infty }(-2/3)^n+\frac{1}{3}\sum _{n=1}^{\infty }(1/75)^n$$