[grey_whiskers]

This sort of sum shows up, for example, in studies of the Gibbs Phenomenon (see Many sine functions for graphs).

Artefacts of an incomplete basis set, then...

Is there any practical interest in either differing convergence rates with the type of discontinuity, the characteristics of the partial series (e.g. if FFT shows different ringing than a conventional Fourier), or the behaviour as you include more and more terms?

...or did I just re-invent a well-known square wheel from the 1800's?

Cheers!

[LjubivojeRadosavljevic]

Is there any practical interest in either differing convergence rates...or the behaviour as you include more and more terms?

Yes, there is a practical interest, it's called ERROR in the approximation of an infinite series. ;-)

[snarks_when_bored]

Is there any practical interest in either differing convergence rates with the type of discontinuity, the characteristics of the partial series (e.g. if FFT shows different ringing than a conventional Fourier), or the behaviour as you include more and more terms?

The Wikipedia entry on the Gibbs phenomenon is pretty informative, g_w. As for practicalities, the entry points out that there's no overshoot/undershoot if wavelet transforms are used. Something else to study (tentatively scheduled for the spring of 2093)...