<s>[INST] For which real $a$ does$ \sum_{n=1}^{\infty} \left\{ e-\left(1+\frac{1}{n}\right)^{n+a} \right\}$ converge or diverge? For which real $a$ does
$ \sum_{n=1}^{\infty} \left\{ e-\left(1+\frac{1}{n}\right)^{n+a} \right\}$ 
converge or diverge?
This is a generalization of
$\sum_{n=1}^{\infty} \left\{ e-(1+\frac{1}{n})^n \right\}$. is this converge or diverge
where it is shown that
the sum diverges for
$a=0$.
My conjecture is that
the sum converges for
$a = \frac12$
and diverges for
$a \ne \frac12$.
 [/INST] I think you are correct.  
Note that
$$\begin{align}\log \left(1 + \frac{1}{n} \right)^{n +a} &= \frac{n+a}{n+1/2}(n + 1/2)\log\left(1 + \frac{1}{n} \right)\\&=\frac{n+a}{n+1/2}(n + 1/2)\left(\frac{1}{n} - \frac{1}{2n^2} +\mathcal{O}\left(\frac{1}{n^3} \right) \right)  \\ &= \frac{n+a}{n+1/2}\left(1 + \mathcal{O}\left(\frac{1}{n^2} \right) \right) \\ &= \left(1 + \frac{a - 1/2}{n + 1/2} \right)\left(1 + \mathcal{O}\left(\frac{1}{n^2} \right) \right)\end{align}$$
We have convergence if $a = 1/2$ and divergence otherwise, since
$$e- \left(1 + \frac{1}{n} \right)^{n +a}= e - \exp \left(1 + \frac{a - 1/2}{n + 1/2} +  \mathcal{O}\left(\frac{1}{n^2}\right)\right) \\ = e\frac{ 1/2-a}{n + 1/2} + \mathcal{O}\left(\frac{1}{n^2} \right)$$
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