Mathematics

Problem

For a real parameter

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.

Intuitive explanation

This series asks you to understand how the exponent $k$ in the denominator controls whether the series converges or diverges—a classic application of the $p$ -series test. The key insight is to rewrite the general term as $\frac{n}{n^k} = n^{1-k}$ and recognize that you're really asking: for which values of $k$ does $\sum n^{1-k}$ converge? Once you identify the critical exponent, you'll see exactly where the boundary between convergence and divergence lies.

Intuitive explanation

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