Mathematics

Problem

Let

p

be a real parameter and consider the series

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

. (a) Compute

\displaystyle \lim_{n\to\infty} \dfrac{1}{n^{p}+1}

as an explicit function of

p

. (b) Determine all real values of

p

for which the series cannot converge. Show analytic work supporting your conclusion. (c) For the remaining values of

p

, explain why knowledge of the limit alone is insufficient to decide convergence, and briefly describe what additional analysis would be needed.