Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero).
Fourier Series: Represents a function over an entire interval (typically $[- L, L]$ or $2\pi]$ ).

Key Differences Between Taylor Series and Fourier Series

Aspect	Taylor Series	Fourier Series
Nature of Expansion	Uses polynomials from derivatives at a single point.	Uses sines and cosines (or complex exponentials) over an interval.
Convergence Conditions	Requires infinite differentiability and a valid radius of convergence.	Requires periodicity and Dirichlet conditions for convergence.
Domain of Representation	Local approximation around a single point.	Represents a function over an entire interval.
Basis Functions	Powers of (x – a).	Sinusoids (sines and cosines) or complex exponentials.
Handling of Discontinuities	Poor handling; requires smoothness.	Can approximate discontinuous functions (with Gibbs phenomenon).
Applications	Local function approximation, differential equations, numerical analysis.	Signal processing, wave analysis, heat conduction, quantum mechanics.

Taylor Series versus Fourier Series for a function