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Math Help - Calculating a series

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Calculating a series

    I came across a problem that asked whether  \sum_{n=1}^\infty \sin(\pi\sqrt{n^2+1}) converges or not. I was able to show that it converges (ask if you'd like to see why), but I was wanting to take this a step further:

    What does  \sum_{n=1}^\infty \sin(\pi\sqrt{n^2+1}) converge to? I'm under the impression that there might not be an answer expressible in terms of finite composition of elementary functions but I'm not sure.

    Thanks everybody
    -Chip
    Last edited by chiph588@; April 23rd 2010 at 01:39 PM.
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  2. #2
    Junior Member
    Joined
    Jan 2009
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    Well my guess would be to look for a fourier series, of a general term:
    a_n cos(nx), where a_n equals your sine function, then I need to solve:
    0=\int_{-\pi}^{\pi} sin(nx) f(x) dx
    and a_n= \int_{-\pi}^{\pi} cos(nx)f(x) dx/N where N is the normalization constant, you could try using fourier transform as in:
    g(n)=\int f(x) exp(-inx)dx (normalization factor is missing) from the above you can find what is g(n) and by inverse fourier transform you can find what is f. cause f(x)=\int g(n)exp(inx)dn...

    This is theoretical, and maybe wrong approach, but that's what came to my mind, didn't do the calculations myself.
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