Comparing Series Convergence: Limit or Direct Comparison Test?

[waealu]

Homework Statement

Does the following series converge or diverge (use either the Limit Comparison or the Direct Comparison Test):

$$\sum_{n=1}^{+\infty} \frac{3^{n-1}+1}{3^{n}}$$

Homework Equations

In a previous problem that was
$$\sum_{n=1}^{+\infty} \frac{1}{3^{n-1}+1}$$
I was able to reindex the series to make it
$$\sum_{n=0}^{+\infty} \frac{1}{3^{n}+1}$$
From there, I took 1/(3ⁿ+1)<1/3ⁿ.

Therefore, since 1/3ⁿ converges, $$\sum_{n=1}^{+\infty} a_{n}$$ also converges.

The Attempt at a Solution

However, I don't know how to solve the series that I'm currently on.

Thanks,
Erik

 

[Dick]

(3^(n-1)+1)/3^n>1/3, isn't it?