Consider the infinite series

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

\displaystyle \sum_{n=1}^{\infty} \dfrac{2^{n}}{3^{n}+n}

. Respond fully to each of the following. (a) Evaluate

\displaystyle\lim_{n\to\infty}\dfrac{2^{n}}{3^{n}+n}

. (b) Based on your result in part (a), determine whether the series converges or diverges. Support your answer with appropriate reasoning.