Suppose a periodic function, $f(t)$ with period $2\pi$ has a Fourier series of
$\sum_{k=-\infty}^{\infty} c_ke^{ikt}$
Now suppose we time shift the function to obtain
$g(t) = f(t-t_0)$
My question is: what effect does this have on the Fourier series? Logically, and thinking about the graph, the series should just shift too, i.e. the Fourier series for $g(t)$ should be given by
$\sum_{k=-\infty}^{\infty} c_ke^{ik(t-t_0)}$
My problem with this chain of thought is that this makes the coefficients decrease much faster, implying a greater rate of convergence. Rearranging the above we obtain:
$\sum_{k=-\infty}^{\infty} c_ke^{-ikt_0}e^{ikt} = \sum_{k=-\infty}^{\infty} d_ke^{ikt}$ for some new set of coefficients $d_k\in\mathbb{C}$.
This makes no sense, since it implies if we just keep shifting the series by $2\pi$, we'd get the same graph, but it would converge much faster. So therefore finding the new Fourier series is more involved that just applying the same time transformation?