Why should we use the Fourier Transform? I'm a CS/Math double major, and during my study (and reading sources out of my own interest) I've had some encounters with the Fourier Transform.
I understand the theory behind Fourier series, and why the integral to calculate the coefficients works. (Nevertheless, my understanding is obviously far from complete; I feel the topics that I ask about are kind of basic).
I basically understand/view the Fourier transform as an extension from the calculations from the discrete coefficients to the real numbers. Instead of a coefficient for each natural number, we have a coefficient for each positive real number.
I have some questions about this approach.
My first question is: why does the Fourier transform make sense? How do we know the Fourier transform spikes at the right spots and not at spots inbetween natural numbers?
Further, the Fourier transform obviously seems have some nice properties, otherwise people wouldn't use it so much. What are the cons of using the Fourier transform as opposed to using the Fourier series?

Short answer: you use Fourier series for functions defined on a compact interval $[a,b]$, and use Fourier transforms for functions defined on the whole line $(-\infty,\infty)$.
Behind the scenes there is some deep theory. Both Fourier series and Fourier transforms are instances of harmonic analysis on a locally compact group: Fourier series on the compact group $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ (the unit circle), and Fourier transforms on the non compact group $\mathbb{R}$.
