I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this.
$$e^{3i\pi/2}+2e^{3i\pi/2}+3e^{3i\pi/2}+4e^{3i\pi/2}+5e^{3i\pi/2}+\cdots$$
Any ideas? If changes to this series are necessary to make it a Fourier series from which an equation is recoverable then that also would be really interesting.
Thanks for the help, much appreciated.
You can factorise $e^{3i\pi/2}$ out of this expression, to get $e^{3i\pi/2}(1+2+3+4+5+\dots)$. So it's divergent (undefined, if you prefer).
It's not really a fourier series anyway: a fourier series is a function. Perhaps you're thinking of something like $f(x) = \dots + a_{-1}e^{-ix} + a_0 + a_1e^{ix} + a_2e^{2ix} + \dots$. Whenever you choose values for the $a_i$, though, you still have to make sure the function converges - that is, you can't take them all equal to $1$, because at $x = 0$ you get $f(0) = \dots + 1 + 1 + 1 + 1 + \dots$, which diverges.