Intuition behind alternate expression of impulse train So it's known that $\sum_n \delta(x-nT) = \frac{1}{T}\sum_m e^{2\pi imx/T}$. This can be proven by expressing the left hand as a Fourier series and finding $c_m$. But it's just mindboggling that this is the case, is it not? Why would one expect this to happen?
If I were to try to plot a crazy amount of the right hand terms, i.e. $cos(2\pi x/T) + i \sin (2\pi x/T)$, would the results actually converge towards something that has value $T$ at all integer multiples of $T$ and zero otherwise? I find it extremely hard to believe. How do you interpret this weird expression? Is there any way to justify it besides the formal proof?

just play with this interactive Desmos Graph and see.
The point is constructive and destructive interferences : every cos contributes around the Dirac comb support or progressively cancel each other at other places, resulting into the Dirac comb.
