Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{1}{n^{1/3} + 2^n}$$
We compare the given series with the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n }$, which is a convergent series ( geometric with $|r|=1/2$) and for $n\geq1$
$$ \frac{1}{n^{1/3} + 2^n}\leq\frac{1}{2^n} $$
Then $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1/3} + 2^n}$ also converges.
