For let and . A student tries to decide whether converges by checking the…

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

Problem

For

n\ge 1

let

a_n=\dfrac{2n^{2}+1}{n^{3}+5}

and

b_n=\dfrac{1}{n}

. A student tries to decide whether

\displaystyle\sum_{n=1}^{\infty} a_n

converges by checking the inequalities

a_n\le b_n

and

a_n\ge b_n

for large

n

, but finds that neither inequality holds eventually. Which of the following best explains why a different comparison can still classify the series

\displaystyle\sum_{n=1}^{\infty} a_n

? (a) The sign of

a_n

alternates while

b_n

is always positive, so inequalities fail but an alternating–series argument succeeds. (b) Because

\displaystyle\sum a_n

can be viewed as the

p

-series

\sum \dfrac{1}{n^{2}}

with

p=2>1

, it converges regardless of any comparison with

b_n

.},{ (c) The ratio

\dfrac{a_n}{b_n}

approaches a non-zero finite limit, so the two series share the same long-term behavior even without a fixed inequality. (d) Because

b_n

does not tend to

0

, direct inequality tests are invalid, but another test works even though

\sum b_n

diverges.