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I want to show $\dfrac{1}{\pi}\sum_{m=1}^{\infty}\dfrac{\sin(2\pi mx)}{m}= -(x-[x]-\dfrac{1}{2})$,when x is not an integer,where [x] denotes greatest integer function . Could you please give any suggestion how to approach?? Thanks

Hitendra Kumar
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    What do you know about Fourier series? – Phira Jul 13 '18 at 04:44
  • Is it related to Poisson Summation formula? I can't see any connection with fourier series. – Hitendra Kumar Jul 13 '18 at 04:57
  • @HitendraTendutolian The left hand side is a Fourier series, in that it's an infinite series of terms from the usual Fourier basis. Try computing the Fourier series for $-(x - [x] - 1/2)$, and see if it matches the left. Then, the Fourier series and the function should agree whenever the function is continuous, i.e. when $x$ is not an integer. – Theo Bendit Jul 13 '18 at 05:06
  • okay,got it. thanks.I am gonna try .If I stuck somewhere I will ask. – Hitendra Kumar Jul 13 '18 at 05:09

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