Can someone help with an improper integral problem

my problem asks to use integration by parts(and L'Hopital's Rule) to show whether the following improper integral diverges or converges. Evaluate the integral if it converges.
Here is the problem and the graph:
http://postimage.org/image/58r2vv7uz/

Could really use someone's help in figuring out how to solve this

Let u = x, dv = e^(-x/2) dx
du = 1 dx, v = -2e^(-x/2).

So, ∫(x = 0 to ∞) xe^(-x/2) dx
= lim(t→∞) ∫(x = 0 to t) xe^(-x/2) dx
= lim(t→∞) [-2xe^(-x/2) {for x = 0 to t} - ∫(x = 0 to t) -2e^(-x/2) dx]
= lim(t→∞) [-2xe^(-x/2) - 4e^(-x/2)] {for x = 0 to t}
= lim(t→∞) -2(x + 2)/e^(x/2) {for x = 0 to t}
= [lim(t→∞) -2(t + 2)/e^(t/2)] + 4
= [lim(t→∞) -2/((1/2)e^(t/2))] + 4, by L'Hopital's Rule
= 0 + 4
= 4.

I hope this helps!