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I asked this question in the math site but haven't got an answer yet, thought I would try here.

I've been revising a few lectures here: https://youtu.be/WgaNe3euf4s?t=9m6s and at the time stamped point in the link, the professor asks to prove an interesting result.

Specifically,

\begin{align} &A_{0}\cos(\Omega_0t+\phi_0)\\ +&A_{0}\cos((2\pi/T\pm \Omega_0)t\pm\phi_0)\\ +&A_{0}\cos((2\times2\pi/T\pm \Omega_0)t\pm\phi_0)\\ &\ldots &= 0 \end{align}

when $t \ne nT$ for integer n.

For $t = nT$, the sum does diverge and seems easy to prove, but for the non-sampled points, I can't seem to show it. I've tried applying trig identities on the individual terms and I managed to get something like

$1+2A_{0}\cos(2\pi/T)t+2A_{0}\cos(2\times2\pi/T)t+\dots$

but I don't know how to proceed. I feel like I need to bring in complex exponentials and find some geometric series. But not sure.

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  • $\begingroup$ Try to write your equation in summation notation. From there, it should be pretty straightforward to get to the answer provided. $\endgroup$
    – Baddioes
    Commented Oct 28 at 19:09

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$$ \sum\limits_{n=-\infty}^{\infty} \delta(t-n) = \sum\limits_{k=-\infty}^{\infty} e^{j 2 \pi k t} $$

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