What Are the Fourier Coefficients for the Half-Range Cosine Series of f(x)=x?

[v_pino]

Homework Statement

The function f(x) is defined on the interval 0<x<L by f(x)=x. It can be represented by the Fourier cosine series

f(x) = a_0 + sum a_n cos(n*pi*x / L)

Find its Fourier coefficients a_0 and a_n.

Homework Equations

Multiply both sides by cos(n*pi*x / L) and integrate from L to 0. Then integration by parts.

The Attempt at a Solution

I got a_0 = L and a_n = 4L/ n^2 pi^2

The answers should be : a_0 = L/4 and a_n = -2L/n^2 pi^2 for n = odd and a_n=0 for n = even.

thanks!

 

[betel]

The give answer is not quite correct, but neither is yours.
a_0 should be L/2 and a_n(odd) = -4L/(n^2 pi^2).
The value for a_0 can be seen to be correct by shifting the function by L/2 to the left. Then it is an odd function up to the shift upwards by L/2. So it can be represented by only sines, which explains that even values of the cosine are 0.

If you want us to find your mistake you have to post your calculation.