Consider the series S= _ n=1 ^ (-1)^ n+1 n

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series

S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}.

A new series

R

is formed by first listing all positive terms of

S

in their natural order and then listing all negative terms in their natural order:

R=1+\frac1{\sqrt{3}}+\frac1{\sqrt{5}}+\cdots-\frac1{\sqrt{2}}-\frac1{\sqrt{4}}-\frac1{\sqrt{6}}-\cdots.

(a) Show that

S

converges but

\displaystyle\sum_{n=1}^{\infty} \bigl|\tfrac{(-1)^{n+1}}{\sqrt n}\bigr|

diverges. (b) Prove that the sequence of partial sums of

R

is unbounded above. (c) Conclude that

R

diverges to

+\infty

. (d) Briefly discuss how the behavior of

R

illustrates a key distinction between absolute and conditional convergence.