In derivation of fourier transform from fourier series coefficient presented here the first step is to multiply the fourier series coefficient by $T$. Could anyone explain why? What's the purpose of doing it?
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A few lines below, I read: $$x(t)=\sum_{n=-\infty}^\infty c_ne^{jn\omega t}=\sum_{n=-\infty}^\infty Tc_ne^{jn\omega t}\frac{1}{T}.$$ He wants to transform a non-periodic function, meaning $T\to\infty$. He sees he can easily calculate the limits of $Tc_n$ and $\frac{1}{T}$ as $T\to\infty$, so he goes ahead. The idea behind this is that $c_n$ is approaching 0, so calculating the limit of $c_n$ directly doesn't lead us anywhere. Instead, $Tc_n$ has a finite nonzero value, and $\frac{1}{T}$ becomes our differential. The sum becomes an integral in $\mathrm{d}\omega$ because sums often have integrals as limits: the integral itself can be seen as a limit of Riemann sums.
In summary, he wants to isolate something finite depending on an integration variable, and the differential of that variable. $\frac{1}{T}$ becomes that differential. The original expression of the sum doesn't have $\frac{1}{T}$, so you have to produce it. You multiply and divide by $T$, so on one side you get the differential, and the other side is precisely $Tc_n$. So there you go: you must calculate $Tc_n$ and its limit.
MickG
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