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I would like to know if there a closed form solution for the sum:

$$ S_n(t) = \sum_{k=0}^{n} \cos( t \sqrt{k} ) $$

There is obviously an easy answer when the sum is replaced by an integral so this question is really asking for the exact form that $S_n(t)$ takes.

Any hints or references on how to approach this problem would be welcome.

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    Have you tried a Taylor expansion? The coefficients should be readily found.2011-01-26
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    @Raskolnikov: I get $ d^{2v} S_n(0) / dt^{2v} = (-1)^v \sum_{k=0}^n k^v $ (odd values of $v$ produce $\sin$ terms which zero out when evaluated at 0), which means $S_n(x) = \sum_{m=0}^{\infty} (-1)^m H_{n,m} x^m / m! $, where $H_{n,m}$ is the harmonic number in $n$ and $m$. WolframAlpha doesn't know how to evaluate this (nor do I). Any suggestions?2011-01-26
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    What would suggest to you that a closed form exists?2011-01-26
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    If the Taylor series expansion is not known by WolframAlpha, chances are that there is no closed form expression. There are formulas for the harmonic numbers, maybe they can help, but I guess they will at best allow you to reexpress $S_n(t)$ in terms of other sums of functions.2011-01-26
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    @Qiaochu Yuan: Is there a way to tell if a closed form exists? If one does exist, is there a way to guess at the form or the constants involved?2011-01-27
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    @user4143: there is an easy way to tell if a closed form exists: find one. However, there is no easy way to tell if a closed form does _not_ exist. There are ways, I think, but they are hard, and I am not familiar with them, and they probably don't always work. The best you can do is probably to show that if a closed form exists then it has bad properties.2011-01-27
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    I've yet to see anything whose closed form involves a square root as an exponent in a series...2011-04-07
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    Terminological nitpick - one does not speak of "solution for the sum," rather, of "evaluation for the sum."2011-05-07

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