Let be the series .

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

Problem

Let

S

be the series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}\, \frac{n!}{(2n)^n}

. (1) Determine whether

S

converges absolutely, converges conditionally, or diverges, and justify your conclusion. (2) State whether the numerical sum of

S

would change if the terms were written in reverse order. Provide a justification based on your result from part (a).