What is value of Pi

f(x)= 0 - L

Question

What is value of Pi

f(x)=    0                 - L<x<0
            L                    0<x<L
fourier series

$$f(x)=\frac{L}{2}+\frac{2}{\pi }\underset{n=1,3,5}{\overset{\infty }{\sum  }} \frac{1}{n}\text{Sin}\left[\frac{(n \pi  x)}{L}\right]$$

$$f\left(\frac{L}{2}\right)=L$$

$$L=\frac{L}{2}+\frac{2}{\pi }\underset{n=1,3,5}{\overset{\infty }{\sum  }} \frac{1}{n}\text{Sin}\left[\left(\frac{1}{2}n \pi  \right)\right]$$

$$1=\frac{1}{2}+\frac{2}{\pi }\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\text{...}\text{..}\right)$$

$$\frac{\pi }{4}=\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\tex$$

anonymous · Answer

$$\frac{\pi }{4}=\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\text{...}\text{..}\right)$$

pokemon23 · Answer

I give up I don't know how to do that?

anonymous · Answer

$$\pi =4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\text{...}\text{..}\right)$$

anonymous · Answer

$$\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\text{...}\text{..}\right)=\underset{n=0}{\overset{\infty }{\sum  }} (-1)^n \frac{1}{2n+1}$$

anonymous · Answer

There you have it, exact value of Pi
$$4\sum _{n=0}^{\infty } (-1)^n\frac{1}{2n+1}=\pi$$

pokemon23 · Answer

ok what grade do i learn this

anonymous · Answer

should have learned it in daycare

pokemon23 · Answer

what! didn't go to day care Darn I missed the freaking lesson

turingtest · Answer

very interesting, but where did the original series come from?

anonymous · Answer

That's fourier series representation of 
this piecewise function
f(x)=    0                 - L<x<0
            L                    0<x<L

turingtest · Answer

oh, is that all?
I need to review Fourier series.
*bookmark

anonymous · Answer

@bahrom7893

bahrom7893 · Answer

dude i suck at series, and calc 2 and 3 in general.

bahrom7893 · Answer

MATH PROCESSING ERROR

anonymous · Answer

just playing