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$\displaystyle{\sum_{n=1}^{\infty} \left[\left( 1 + {1 \over n}\right)^{n} - {\rm e}\right]}$

I tried both root and ratio tests (for the root test, the expression became way too complex to handle) but both didn't really work out.

Would there be an easy way of showing that this diverges?

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    $\begingroup$ see here: math.stackexchange.com/questions/703207/… $\endgroup$ – Eleven-Eleven Mar 12 '14 at 2:38
  • $\begingroup$ Essentially, you need to estimate $(1+1/n)^n-e$. $\endgroup$ – Thomas Andrews Mar 12 '14 at 2:38
  • $\begingroup$ thanks a lot! I didn't know there was already a question about it. $\endgroup$ – user98235 Mar 12 '14 at 2:41
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    $\begingroup$ it popped up a couple of days ago and I was intrigued. So when I saw yours I posted the link. Hope it helps. $\endgroup$ – Eleven-Eleven Mar 12 '14 at 2:43