Problem: Give a simple explanation of why, if the Fourier coefficients $a_{k}=b_{k}=0$ for all sufficiently large $k \gg 1$, then the Fourier series converges to an analytic function.
Attempted Explanation: Since the Fourier series is an infinite sum, if the series did not converge, then the function would blow up. Since $a_{k}=b_{k}=0$, and $k>>1$, the high valued k's finely tune the oscillating function to an analytic function.
This explanation was apparently not good because I only got 0.2/1 credit for the problem, thus I am wondering what a better explanation would be for it.
So there is some $N>0$ such that for all $k>N$ the coefficients are zero, you mean? Then it is easy since you have a finite sum of analytic functions. Finite sums of analytic functions are analytic.