Let S= _ n=1 ^ ^ 2 n n^ 2

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

Problem

Let

S=\sum_{n=1}^{\infty} \frac{\sin^{2} n}{n^{2}}.

Which choice correctly describes the convergence of

S

? (a)

S

diverges because the comparison with

\displaystyle\frac{1}{n}

shows the partial sums grow without bound. (b)

S

converges because each term is non\text{-}negative and is bounded above by

1/n^{2}

, whose series is convergent. (c)

S

converges conditionally but not absolutely since the terms fail to form a monotonically decreasing sequence. (d)

S

converges by the alternating series test because the factor

\sin^{2} n

causes the signs of the terms to oscillate.