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I am curious if there is a closed form for this summation $\sum\limits_{k=0}^{\infty}\sqrt k\cos(kx)$

I am aware that $\sum\limits_{k=0}^{\infty}\cos(kx)$ resembles a dirac delta comb and $\sum\limits_{k=0}^{\infty}k\cos(kx)$ can be expressed as $\;-\dfrac{1}{4}\csc^2 \left(\dfrac x2\right)$.

But I am not able to derive the square root case from those. Any hints appreciated.

Angelo
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Srini
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  • "$\sum\limits_{k=0}^{\infty}k\cos(kx)$ can be expressed as $;-\dfrac{1}{4}\csc^2 \left(\dfrac x2\right)$" Source? That sum clearly diverges (and also yours) – leonbloy May 01 '23 at 18:26
  • Sorry for the lazy answer: https://www.wolframalpha.com/input?i=sum+k%3D0+to+infinity+%28+k+cos%28kx%29%29 I will try to derive it first hand if I can – Srini May 01 '23 at 18:29
  • Well, that's obviously wrong. – leonbloy May 01 '23 at 18:40
  • Not sure I would agree with that. InfiniteSum (sin(kx)) is a scaled cotangent function (see formula 19 here: https://mathworld.wolfram.com/Sine.html). If you differentiate sin(kx) w.r.t x, you get k cos(kx). Similarly if you differentiate cot(x), you get csc^2(x). – Srini May 01 '23 at 18:50
  • Although I have to agree that my knowledge of various types of summations is limited and I might have applied formulas out of context. Thanks for your comments though – Srini May 01 '23 at 18:53
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    One thing to watch out for; whatever the result is, closed-form or not, it will blow-up at every integer multiple of $2\pi$ as $\sum^\infty_{k=0} \sqrt{k}$ diverges. – aghostinthefigures May 01 '23 at 19:05
  • https://math.stackexchange.com/questions/3426075/evaluating-sum-k-0-infty-sinkx-and-sum-k-0-infty-coskx – leonbloy May 01 '23 at 19:06
  • @aghostinthefigures, yes, I expect that. From partial sums, I do see some polylog functions $Li_{-\frac{1}{2}}$(as with infinite sum of cos(kx)/k, although a different polylog). csc blows up as well at multiples of $\pi$. So definitely agreed on the blow up part, but still interested if this approximates a closed form of something that I can recognize – Srini May 01 '23 at 19:21
  • @Srini The type of summation you are considering is necessary context for this question. Without that context specified, the default interpretation for anyone stumbling upon this question would be pointwise convergence, for which the series you've specified are easily shown to diverge. – Brian Moehring May 01 '23 at 19:49
  • The sum could be $0$ if $\cos(kx)=0$, but there is no solution for $x$ given $k\in\Bbb N$. Also, complex $x$ makes the sum diverge too, so likely the sum diverges for all values of $x$. – Тyma Gaidash May 01 '23 at 19:49
  • There's an expression for $\sum_{k=1}^\infty, k^{-1/2}\sin kz$ so it's possible a CAS would proceed formally, differentiating it with respect to $z$ to get the summation you're asking about. – A rural reader May 01 '23 at 20:04
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    @Aruralreader, your hint is helpful and is what I was looking for. I do note the caution others have asked me to exercise interpreting the sum. – Srini May 01 '23 at 21:45

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