Mathematics

Problem

Analyze the series

\sum_{n=1}^{\infty} \frac{n}{n^{3/2}+\sin^{2} n}.

Which statement about its convergence is correct? a) The series converges absolutely by the limit comparison test with

\displaystyle\sum_{n=1}^{\infty}\frac1{n^{3/2}}

.},{ b) The series diverges by comparison with the divergent

p

-series

\displaystyle\sum_{n=1}^{\infty}\frac1{\sqrt n}

. c) The series converges conditionally because the

n

in the numerator changes sign while the absolute series diverges. d) The integral test shows convergence since

\displaystyle\int_1^{\infty}\!\frac{dx}{\sqrt{x}}=2