Let a_n= n^ 2 +1 n^ 3 -n (n 2), S= _ n=2 ^ a_n

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

a_n=\frac{n^{2}+1}{n^{3}-n}\qquad (n\ge 2),\qquad \quad S=\sum_{n=2}^{\infty}a_n.

Assume standard facts about

p

-series. (a) Choose an elementary benchmark series whose general term is of the form

c/n^{p}

(

c>0

) that will allow you to decide whether

S

converges or diverges. Explain why this choice is appropriate. (b) Compute

\displaystyle \lim_{n\to\infty}\frac{a_n}{b_n}

, where

b_n

is the general term of your benchmark from part (a). Show all work leading to the limit. (c) Determine whether the series

S

converges or diverges. Justify your conclusion using the result of part (b) and the known behavior of the benchmark series.