Mathematics

Mathematics

Problem

Consider the infinite series ∑n=1∞2n3n+n\displaystyle \sum_{n=1}^{\infty} \dfrac{2^{n}}{3^{n}+n}. Respond fully to each of the following. (a) Evaluate lim⁡n→∞2n3n+n\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.

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