Question

Is the series `\sum_(n=1)^\infty n e^(-n)` absolutely convergent, conditionally convergent or divergent?

(Use the appropriate test)

Answer 1

Converges

Explanation:

Apply the ratio test:

`lim_(n to oo ) |a_(n+1)/a_n |`

`= lim_(n to oo ) |((n+1) e^(-(n+1)))/(n e^(-n)) |`

`= lim_(n to oo ) |((1+1/n) )/( e) | = 1/e lt 1`

It converges :)