
Calculating a series
I came across a problem that asked whether $\displaystyle \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 $\displaystyle \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

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.