I'm just beginning to learn about Fourier series and I'm trying to figure out how to find the Fourier series coefficients for $x(t) = e^{i100\pi t}$
I know also that $$x(t) = \sum_{-\infty}^{\infty} a_{k} e^{ik(2\pi/T)t}$$
How do I find these coefficients? I'm not even sure where to begin.
We can make the $k$th term in the series $$\sum_{k=-\infty}^{\infty}a_k e^{ik(2\pi /T)t}$$ equal to $e^{i100\pi t}$ if $$k(2\pi/T) = 100\pi$$ or equivalently $$k = 50T$$ Note that if $T$ is the fundamental period of the series. If $T$ is not specified in advance, then there are infinitely many solutions. We require $k$ to be an integer, which will be true as long as $T$ is any integer multiple of $1/50$. So, for any positive integer $n$, the following gives us a perfectly valid representation of $x(t)$ as a Fourier series: $$a_k = \delta(k,n) \text{ and }T = n/50$$ where the $\delta$ is the Kronecker delta: $$\delta(k,n) = \begin{cases}1 & \text{if }k = n \\ 0 & \text{otherwise} \\ \end{cases}$$ We exclude integers $n \leq 0$ because the fundamental period $T = n/50$ should be positive.