Why does this series converge by the integral test? Problem inside.

Question

Why does this series converge by the integral test?  Problem inside.

anonymous · Answer

$$\sum_{n=0}^{\infty}n e ^{-n ^{2}}$$

anonymous · Answer

I keep getting a final answer of $$(-1/2) e^{-b ^{2}} + 1/2 e^{-0^{2}}$$where b is going to infinity. I would think that would come out to infinity plus 1/2, which would be infinity and cause the whole thing to diverge. However, according to problem (d) on this study sheet I found, the answer converges. Something is wrong with my algebra. Help please? http://www.hillsdalesites.org/personal/dmurphy/Kclasses/math113/review2solns.pdf Actually, I think I just solved my own problem. Would it be 0 + 1/2 because the $$^{-b ^{2}} $$ moves e to the denominator and thus makes that part go to zero?