For a real parameter

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

Problem

k

, define the series

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

. (a) Express the general term

a_n(k)

S(k)

in simplest form as a power of

n

. (b) Determine

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

in terms of

k

. (c) For which values of

k

does the limit test guarantee that

S(k)

diverges? Provide complete reasoning, including all intermediate algebraic steps. (d) Identify the values of

k

for which the limit test is inconclusive and briefly describe why no conclusion about convergence can be drawn from the limit test alone in those cases.

Related theory

Infinite Series
Maclaurin Series

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