Series expansion of {x}

Question

Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to prove it and know why does that happen.
$$\{x\} = \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2k\pi x)}{k}$$ This is the original link. Thank you very much

Answer 1

If you were to plot a graph of the fractional part of $x$ against $x$, you would get essentially a "sawtooth wave", as the fractional part starts at 0 (when $x$ is an integer) and increases linearly to just below 1, until just before the next integer (where the 1 will 'wrap around' to become 0 again). Try plotting this function, either a sketch by hand or use some graphing software. You will see (and I won't prove it here) that $f(x)=\{x\}$ is a periodic function with period 1 (that is it repeats every 1 unit on the $x$-axis). Because of this property it is expressible as a Fourier series. If you were to shift the $x$-axis up by 1/2 a unit, you would have an odd function, which would mean you would only get sine terms in the Fourier series. So, in your formula, the 1/2 comes from the shift of the horizontal axis, and the sine terms constitute the Fourier sine series for the resulting function.

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