For each real number

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

Problem

k

, define the series

T(k)=\sum_{n=1}^{\infty} \frac{k^{\,n}}{n^{2}}.

Respond to the prompts below. (a) Obtain a simplified algebraic expression for

\dfrac{t_{n+1}}{t_n}

, where

t_n=\dfrac{k^{\,n}}{n^{2}}

. Express your answer in terms of

k

and

n

. Do not take any limits in this part. (b) Compute

\displaystyle \lim_{n\to\infty}\dfrac{t_{n+1}}{t_n}

in terms of

k

. Show your reasoning carefully. (c) Using your result from part (b), determine all real values of

k

for which the series

T(k)

converges and all real values for which it diverges. If your work in part (b) is inconclusive for any value(s) of

k

, identify those value(s) and briefly explain why the convergence question remains unresolved for them.