Mathematics

Problem

Consider the series

\displaystyle \sum_{n=1}^{\infty} a_n(k)

with terms

a_n(k)=\dfrac{k^{n}}{n}

, where

k

is a real parameter. (a) Evaluate

L(k)=\displaystyle \lim_{n\to\infty} a_n(k)

in terms of

k

. Clearly state all cases that arise. (b) Using your result from part (a), identify all real numbers

k

for which the series cannot converge. Provide a complete justification based on the behavior of the sequence

\{a_n(k)\}

. (c) When

|k|\le 1

, does the information obtained from part (a) guarantee convergence or divergence of the series? Explain fully, citing the relevant reasoning about the limit you computed.

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