Examples of Taylor SEries versus Fourier Series
Intro
Which works better for a given function – a Taylor expansion or a Fourier Expansion? This post explores the pros and cons of each, using specific examples.
Examples of Taylor and Fourier Series Expansions
1. Polynomial Function: 
Taylor Series Expansion:
Fourier Series Expansion:
Best Fit: Taylor series
2. Trigonometric Function: 
Taylor Series Expansion:
Fourier Series Expansion:
Best Fit: Fourier series
3. Exponential Function: 
Taylor Series Expansion:
Fourier Series Expansion: Not practical
Best Fit: Taylor series
4. Piecewise Function:
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Taylor Series Expansion: Not possible
Fourier Series Expansion:
Best Fit: Fourier series
5. Periodic Step Function: 
Taylor Series Expansion: Not possible
Fourier Series Expansion:
Best Fit: Fourier series
Comparison Table
Function | Taylor Series | Fourier Series | Best Fit |
---|---|---|---|
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Good (converges well) | Works if periodic but inefficient | Taylor series |
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Good for small ![]() |
Best for periodic representation | Fourier series |
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Excellent (globally convergent) | Poor (unless forced periodicity) | Taylor series |
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Not possible | Works well (some Gibbs effect) | Fourier series |
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Not possible | Best option (Gibbs phenomenon) | Fourier series |