Mathematics

Problem

Consider the series with general term

d_n = \dfrac{n}{n^{4}+1}

for

n \ge 1

. (a) Compute the improper integral

\displaystyle L=\int_{1}^{\infty} \frac{x}{x^{4}+1}\,dx

, showing each step of your calculation. (b) Using your result from part (a), determine whether the series

\displaystyle \sum_{n=1}^{\infty} d_n

converges or diverges. Provide a clear justification.