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.