I am trying to understand the conversion of Fourier series to the Fourier integral. The book I am referring to is "Advanced Engineering Mathematics" by Erwin Kreyszig. The proof contains the following style of conversion:
$ \lim_{\Delta \alpha\rightarrow 0} \sum_{n=1}^{\infty} F(\alpha_n)\Delta \alpha \rightarrow \int_{0}^{\infty}F(\alpha)d\alpha $
How is that summation under zero-approaching limit converted to an improper integral ? What's the essential piece I am missing here to understand it?
We have that $\Delta\alpha=\alpha_n-\alpha_{n-1}$. As $\Delta\alpha\to0$, the $\alpha_n$ get closer together, so you approach a summation over infinitesimally small intervals. Notice that $F(\alpha_n)$ is the height of the $n$th rectangle. So, multiplying it by the width of that interval, that is $\Delta\alpha$, you get the area of that rectangle. We know integrals are defined in this way, so you actually get the integral $\int_0^\infty F(\alpha)d\alpha$, which measures the area under the curve $y=F(\alpha)$.