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$$s_n = \sum_{k=1}^n e^{1/k}$$

This sum came up while practicing closed-form finding on a calculus book's infinite series chapters.

Using Concrete Mathematics' perturbation method, I arrive at

$$s_n + e^{1/k+1} = e + \sum_{k=1}^n e^{1/k+1}$$

Which I can't seem to rewrite with a constant in front of the sigma. I noticed a lot of the sums have rational powers with the index in the denominator. In general, is there a way to solve these rational sums?

Charles
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  • Well, there is no nice analytic representation of your sum. That is obvious. So what do you want? If $k$ is a large number, then $exp(1/k)$ is close to $1$. You could expand the exponential as $1 + 1/k+ ....$. – M. Wind Jun 15 '15 at 17:59
  • I wondered if there was a closed-form for the partial sums. You say it's "obvious" that none exists; could you say why? (of course the sum diverges, but that doesn't mean there isn't a nice expression for it) – Charles Jun 15 '15 at 18:23

1 Answers1

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There are no known closed forms for sums of the form $\displaystyle\sum_na^{n^b}$, for $a\not\in\big\{0,\pm1\big\}$ and $b\not\in\big\{0,1,2\big\}$ In our case, $a=e$, and $b=-1$.

Lucian
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