Understanding of the fourier coefficients?

[cabellos]

Im just looking through examples of finding the Fourier coefficients.

in one particular example bn is found to be = (-1/npi) (cosnpi - cos 0)

then it says this is 0 when n is even
and 2/npi when n is odd

are we just substituting values of n e.g. 1,2,3... to find this result?

i apologise if i am doing something stupid with my calculation but when i put even numbers in place of n i do not get 0 as it states i should?

 

[Päällikkö]

cos(n*pi) - cos(0) = cos(n*pi) - 1 = (-1)ⁿ - 1

(-1)ⁿ = 1 when n is even and -1 when n is odd.
So when n is even: 1-1 = 0, when odd: -1-1 = -2
These are of course divided by -n*pi to get b_n.

 

[HallsofIvy]

cosine of an odd multiple of $\pi$ is -1 and cosine of an even multiple of $\pi$ is 1. That's exactly what they are doing.