For , consider the series . Which statement about its convergence is…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

For

k>0

, consider the series

\displaystyle \sum_{n=1}^{\infty} \frac{k^{\,n}\,n!}{n^{n}}

. Which statement about its convergence is correct? (a) It diverges for every positive value of

k

. (b) It converges if

k\le e

and diverges if

k>e

. (c) It converges if

k<e

and diverges if

k\ge e

. (Correct) (d) It converges if

k<1

and diverges if

k\ge 1