For a real constant

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

Problem

For a real constant

k

, consider the series

\displaystyle\sum_{n=1}^{\infty}\dfrac{k^{n}}{n}

. The behavior of this series depends on the value of

k

. Answer each part thoroughly. (a) Compute

\displaystyle\lim_{n\to\infty}\dfrac{k^{n}}{n}

for the cases

|k|<1

|k|=1

, and

|k|>1

. Express each limit in simplest form or state when it fails to exist. (b) Using the results of part (a), determine all real values of

k

for which the series diverges. Provide a clear justification for every interval or point you identify. (c) Briefly discuss why the analysis in parts (a)–(b) cannot by itself establish convergence when the series does not diverge outright.

Related theory

Infinite Series

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