series from n = 1 to infinity, (ne^(-n))
The Attempt at a Solution
I want to use integral test.
I know this function is:
positive (on interval 1 to infinity)
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??