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## Homework Statement

series from n = 1 to infinity, (ne^(-n))

## Homework Equations

## The Attempt at a Solution

I want to use integral test.

I know this function is:

positive (on interval 1 to infinity)

continous

and finding derivative of f(x) = xe^(-x) I found it to be ultimately decreasing.

So integral test is applicable.

I set up integral from 1 to infinity (xe^(-x))

u = x du = dx

v = -e^(-x) dv = e^(-x)

-xe^(-x) + integral e^(-x)

-xe^(-x) - e^(-x) = -e^(-x) (x + 1)

evaluating from 1 to t

[itex] -(1/e^t) (t+1) + 2/e^(1) [/itex]

but now when I do lim t -> infinity, -(1/e^t) (t+1) should = infinity/infinity, which would mean [itex] a_n [/itex] would be divergent, but it is convergent.

does anyone know where my mistake is??