• Dec 4th 2011, 01:42 AM
mami
It looks to me very much as though this series diverges. In fact, a bit of work with a pocket calculator suggests that $e - \bigl(1+\tfrac1n\bigr)^n > \tfrac1n$ whenever $n\geqslant3.$ More precisely, I'm finding that $n\Bigl(e - \bigl(1+\tfrac1n\bigr)^n\Bigr)$ converges to a limit approximately 1.36 as $n\to\infty.$
The series has non-negative terms, and $e-\left(1+\frac 1n\right)^n\sim\frac e{2n}$ hence the series diverges, hence the sum is $+\infty$.