I was working on the Furie series and for a moment I saw the connectivity between these series and fractals. An example of this can be seen by visualizing an approximation of a square wave by taking the first 1, 2, 3 and 4 terms of its Fourier series.
Simone Orcioni
What you mention, it is the well known Gibbs phenomenon.
https://en.wikipedia.org/wiki/Gibbs_phenomenon
I'm not an expert in fractals, but it seems that in this case both recursivity and self-similarity are missing.
The "informal" explanation of wikipedia undelines better the difference : "Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sine and cosine waves."
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