Determine whether the series converges or diverges.

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

Problem

Determine whether the series

\displaystyle S=\sum_{n=1}^{\infty}\frac{3}{n^{2}+1}

converges or diverges. (a)

S

diverges because its terms behave like

\dfrac{3}{n}

for large

n

, a divergent

p

\,-series. (b)

S

converges because its terms are eventually less than

\dfrac{3}{n^{2}}

, a convergent

p

\,-series. (c)

S

converges because its terms form a geometric sequence with ratio less than

1

. (d)

S

diverges because the denominator grows more slowly than

n^{2}

for large

n