Answer
Converges
Work Step by Step
Given
$$ \sum_{n=1}^{\infty}\frac{2}{3^{n}+3^{-n}} $$
Compare with $\displaystyle\sum_{n=1}^{\infty}\frac{2}{3^{n}} $, which is a convergent series ( geometric with $|r|<1$) and for $n\geq 1$
\begin{align*}
3^{n}+3^{-n} &\geq 3^{n}\\
\frac{1}{3^{n}+3^{-n}} &\leq \frac{1}{3^{n}}\\
\frac{2}{3^{n}+3^{-n}} &\leq \frac{2}{3^{n}}
\end{align*}
Then $\displaystyle\sum_{n=1}^{\infty}\frac{2}{3^{n}+3^{-n}} $ also converges.
