Mathematics

Problem

The series

\displaystyle S=\sum_{n=1}^{\infty} n e^{-n}

is given. Let

h(x)=x e^{-x}

for

x\ge 1

. (a) Verify that

h(x)>0

and continuous for all

x\ge 1

. Explain your reasoning. (b) Show analytically that

h

is decreasing on

[1,\infty)

. (c) Evaluate the improper integral

\displaystyle \int_{1}^{\infty} x e^{-x}\,dx

. Show all steps leading to your result. (d) Decide whether the series

S

converges or diverges. Justify your conclusion using the work from parts (a)–(c).