Tools / Series
Series And Fourier Lab
Compare a user-entered function against its local Taylor approximation and a Fourier partial sum. This lab is built to help students and engineers see where local power-series reasoning and global periodic approximation agree, and where they do not.
Approximation Summary
Fourier Coefficients
Taylor Coefficients
Function Versus Approximations
The full chart is shown on [-pi, pi]. The Fourier partial sum is meant to approximate the
function over the whole interval, while the Taylor polynomial is only expected to behave well near the chosen center.
How To Read The Output
Taylor coefficients tell you about local behavior at one point. Fourier coefficients tell you how much of each sine and cosine mode is needed to reconstruct a periodic signal over a wider interval.
- Increase Taylor order to improve the local approximation near the center.
- Increase Fourier harmonics to capture more oscillatory structure across the whole window.
- Try
abs(x)or a sharper waveform to see the difference between smooth and non-smooth behavior.
How To Use This Lab
Enter a function of x, then choose how many Fourier modes and how many Taylor terms to keep.
The chart compares the original function with both approximations at once.
- Use the Taylor controls to study local approximation near one point.
- Use the Fourier controls to study periodic approximation across a whole interval.
- Compare the coefficient panels to see what each approximation is actually encoding.
Fundamental Mathematics
A Taylor series approximates a function near a chosen center using derivatives at that point. A Fourier series approximates a periodic function using weighted sine and cosine modes over an interval. These are different ideas: one is local and derivative-based, the other is global and frequency-based.
Putting them side by side helps show why smooth local behavior does not automatically imply a good global approximation, and why oscillatory structure often becomes clearer in a Fourier basis.