[nextaccountic]

the fourier transform of a periodic signal is composed of a train of dirac deltas, each multiplied by some factor

the delta with smallest frequency is the fundamental frequency, and the others are harmonics

when you do the inverse fourier transform on this train, each delta becomes a sinusoid

that's how you can write any periodic function as a sum of sinusoids, all of them multiples of the fundamental frequency

and that's the fourier series: it's just the fourier transform, followed by an inverse fourier transform, macroexpanded

but the fourier series only work for periodic functions, because only periodic functions have a bunch of isolated, periodic deltas as its fourier transform

so the fourier transform is only half the step of a fourier series (to write down the series you also need the inverse fourier transform) but, at the same time, the fourier transform is a generalization of the fourier series, because it works for nonperiodic functions too

[nextaccountic]

To elaborate more, when we say that a signal is the linear combination of infinitely many frequencies, if those frequencies are multiples of a fundamental frequency (that is, a countable set of frequencies), then we are talking about a Fourier series and the signal must be periodic

But if those frequencies span a whole continuum (rather than frequencies f, 2f, 3f, 4f..), that is, an uncountable set of frequencies, then this signal is non-periodic and we can't talk about a Fourier series anymore, we must use the Fourier transform

[BeetleB]

> the delta with smallest frequency is the fundamental frequency, and the others are harmonics

Nitpick, but this isn't true. If my signal is a linear combination of two sinusoids - one at frequency 3 and the other at frequency 5, then there is no "fundamental" frequency when you do the FT.