# Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series

Consider the function $f(x) = \frac{x}{2}$, defined over the interval $[0, 2\pi]$. Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.

• I think you might have typoed. The function $f(x)=\frac{x}{2}$ does not have a period. Commented Dec 18, 2015 at 12:47
• Assuming the case that it has a period 2pi Commented Dec 18, 2015 at 12:50
• So do you mean: $$f(x)=\pi\left(\frac{x}{2\pi}-\bigg\lfloor\frac{x}{2\pi}\bigg\rfloor\right)$$ Commented Dec 18, 2015 at 12:54
• You can keep the function to be f(x) = x/2. This is one of the questions asked in my quiz yesterday. I think you would know better what we can do with this and how this can be shown. Commented Dec 18, 2015 at 12:57
• Hint: You must have done Fourier series. Commented Dec 18, 2015 at 13:00

You just have to expand $y=x/2$ as a Fourier series: $${x\over2}=\sin (x)-\frac{1}{2} \sin (2 x)+\frac{1}{3} \sin (3 x)-\frac{1}{4} \sin (4x)+\frac{1}{5} \sin (5 x)+\ldots$$ and put here $x=\pi/2$.
• As a matter of fact the question is not well formulated: $f(x)$ should be defined as $x/2$ on $[-\pi,\pi]$ and extended by periodicity for the other values of $x$. Commented Dec 18, 2015 at 14:43