Analyze the series _ n=1 ^ n n^ 3/2 + ^ 2 n

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Analyze the series ∑n=1∞nn3/2+sin⁡2n.\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 ∑n=1∞1n3/2\displaystyle\sum_{n=1}^{\infty}\frac1{n^{3/2}}.},{ (b) The series diverges by comparison with the divergent pp-series ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty}\frac1{\sqrt n}. (c) The series converges conditionally because the nn in the numerator changes sign while the absolute series diverges. (d) The integral test shows convergence since ∫1∞ ⁣dxx=2\displaystyle\int_1^{\infty}\!\frac{dx}{\sqrt{x}}=2.

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