a problem about series

**mami** · December 4th 2011, 01:42 AM

i tried... i can't get tackle with this problem no matter how hard i try. please help me.

**Opalg** · December 4th 2011, 08:14 AM

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 $n\to\infty.$

**girdav** · December 4th 2011, 08:16 AM

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 $+\infty$ .

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