Analyze the series U= _ n=2 ^ n n^ 2

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

Problem

Analyze the series

U=\sum_{n=2}^{\infty}\frac{\ln n}{n^{2}}.

Answer the questions below. (a) Show that

\ln n < n^{1/2}

for every integer

n\ge 2

. Provide an explicit proof (you may use calculus or inequality manipulation). (b) Use the bound from part (a) to produce an inequality that relates the general term of

U

to a

p

-series term. (c) Determine the convergence or divergence of

U

. Include full justification based on your comparison in part (b). (d) Suppose

V=\sum_{n=2}^{\infty}\dfrac{\ln n}{n^{3/2}}

. Without re-proving part (a), decide whether

V

converges and explain how your reasoning in parts (a)–(c) can be adapted to this new series.

Related theory

Infinite Series
Comparison Tests

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