Suppose is a sequence of positive real numbers and consider the infinite series .

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

Problem

Suppose

\{a_n\}

is a sequence of positive real numbers and consider the infinite series

\displaystyle \sum_{n=1}^{\infty}(-1)^{n+1}a_n

. (1) Explain why the alternating nature of the summand together with the fact that

a_n\to 0

n\to\infty

is not, by itself, sufficient to guarantee that the series converges. (2) Describe the additional property that must be verified for

\{a_n\}

in order for the series to converge, and justify why this property matters. (3) Construct an explicit example of a sequence

\{a_n\}

for which

a_n>0

and

a_n\to 0

, but

\displaystyle \sum_{n=1}^{\infty}(-1)^{n+1}a_n

fails to converge. Support your claim with a brief explanation.