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will the following converge if they then what are their sums

$\sum_{n=1}^\infty\left[\frac{\left(1+\frac{1}{n}\right)}{e}\right]^n$

$\sum_{n=1}^\infty\left[\frac{\left(1+{1}/{n}\right)^{n+1/2}}{e^n}\right]$

do the above series have any physical meaning (apart from fun).

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The partial sums are increasing and are bounded above by $\sum_{n=1}^\infty\left[\frac{\sqrt{8}}{e^n}\right] = \frac{\sqrt{8}}{e-1} \approx 1.646\ldots $ so they converge.

They in fact converge to just under 1.23 and 1.63. I would not expect there to be a physical meaning.

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Hint: You can use the root test to prove the convergence.

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