How Can You Solve Alternating Solutions in Fourier Series?

[musk]

Homework Statement

This is a general question, no real problem statement and is connected to solving Fourier series. You know that to solve it, you need to find a_{n}, a_{0} and b_{n}.

Homework Equations

When solving the above mentioned ''coefficients'' you can get a solution with sin or cos which, in the case of cos(n\pi) can be written as (-1)^{n} where n=0,1,2,... since the solution alternates between 0,1,-1. Is there any similar way to write (or solve) sin or cos\frac{n\pi}{2} since the solution follows this pattern 1,0,-1,0,1,0,-1,...

Thank you in advance!

 

[vela]

Typically you then have only odd or even terms, e.g. $a_k \cos (k\omega t), k=1, 3, 5,\dots$. You can reindex the series using the substitution k=2n+1 or k=2n. The alternating sign is conveniently expressed by (-1)ⁿ or (-1)ⁿ⁺¹.