when I was deriving fourier transform from fourier series, I encoutered a difficulty of thinking in converting the summation into integral (I'm from engineering background). So could someone give some semi-rigorous proof of the following conversion? Cheers
\begin{equation}\lim_{\delta x \rightarrow 0} \sum_{n=-\infty}^\infty f(n\delta x) \delta x =\int_{-\infty}^\infty f(x) dx \end{equation}
The method taught in my notes is that when $T \rightarrow \infty$, the $n\omega_0$ terms in the series (where $\omega_0 = \delta\omega =\frac{2\pi}{T} \rightarrow 0$ and $\,T\,$ is the period of an arbitrary (continuous with infinity range?) function) becomes a new variable $\omega$.
I just don't get how the summation series with index $n$ goes to integral at the same time while a new variable is introduced. Normally what I was taught before is: \begin{equation} \lim_{\delta x \rightarrow 0} \sum_{x=a}^b f(x) \delta x =\int_{a}^b f(x) dx \end{equation}