Fourier Series- half range sine series

[Kamekui]

Homework Statement

Let f(x)=x, 0≤x≤p

(a.) Compute the half-range sine series
(b.) Use the series to show that 1-(1/3)+(1/5)+...=π/4

Homework Equations

b_n=(2/L)*int(from 0 to L) f(x)*sin(nπx/L) dx

The Attempt at a Solution

b_n=(2/p)*int(from 0 to p) x*sin(nπx/p) dx

Using integration by parts I get:

b_n= 2[(-p*cos(nπ)/nπ) + (p*sin(nπ)/n²π²)]

I'm not really sure where to go from here, any help is appreciated.

 

[vela]

Since n is an integer, you can simplify $\cos n\pi$ and $\sin n\pi$.