Consider the series _ n=2 ^ n n^ 2 +1

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

Problem

Consider the series

\sum_{n=2}^{\infty} \frac{\sqrt n}{n^{2}+1}.

Which of the following statements about the series is true? (a) The series diverges because its terms are eventually larger than

1/n

and therefore behave like the harmonic series. (b) The series diverges because the ratio test yields a limit of

1

, which signals divergence. (c) The series converges because its terms are eventually smaller than a multiple of

1/n^{3/2}

, whose series is convergent. (d) The series is alternating and therefore convergent by the alternating–series test.