I'm prette sure the first sum diverges. One can check that [LaTeX Error: Syntax error] for integer [LaTeX Error: Syntax error].
Now, the harmonic series diverges, and so diverges as well.
but since [LaTeX Error: Syntax error] for [LaTeX Error: Syntax error]
then [LaTeX Error: Syntax error]
and so by the comparison test, both sums diverge, and since ,
your series in problem 1 diverges.
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how to solve this problem? 
![\sum_{n=1}^{\infty}\left[e-\left(1+\frac{1}{k}\right)^k\right]=(e-2)+(e-\frac{9}{4})+\sum_{n=3}^{\infty}\left[e-\left(1+\frac{1}{k}\right)^k\right]](http://www.mathhelpforum.com/math-help/latex2/img/8dd7af7ee9a04e6d806c8528f726ff69-1.gif "\sum_{n=1}^{\infty}\left[e-\left(1+\frac{1}{k}\right)^k\right]=(e-2)+(e-\frac{9}{4})+\sum_{n=3}^{\infty}\left[e-\left(1+\frac{1}{k}\right)^k\right]")
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