i tried... i can't get tackle with this problem no matter how hard i try. please help me.
It looks to me very much as though this series diverges. In fact, a bit of work with a pocket calculator suggests that $\displaystyle e - \bigl(1+\tfrac1n\bigr)^n > \tfrac1n$ whenever $\displaystyle n\geqslant3.$ More precisely, I'm finding that $\displaystyle n\Bigl(e - \bigl(1+\tfrac1n\bigr)^n\Bigr)$ converges to a limit approximately 1.36 as $\displaystyle n\to\infty.$