Online Tool Store
Taylor Series Expander
Expand a function as a Taylor or Maclaurin series to a chosen number of terms, with the remainder and the interval where it converges.
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Math & Science
Taylor Series Expander
Frontend preview — no upload or external service.
Series expansion
e^x about 0 to six terms: 1 + x + x²/2 + x³/6 + x⁴/24 + x⁵/120. At x = 0.5 that is accurate to about 2×10⁻⁵; at x = 3 the error is still large.
How the Taylor Series Expander works
- Enter the function and the point to expand about — zero gives the Maclaurin series.
- Choose how many terms; accuracy improves fast near the expansion point and slowly far from it.
- Check the interval of convergence, since outside it more terms make the approximation worse, not better.
The method
A Taylor series represents a function as an infinite sum of its derivatives at a point, each divided by a factorial.
f(x) = Σ fⁿ(a)(x−a)ⁿ / n!
Truncating after n terms leaves a remainder that grows with distance from the expansion point — which is why the same six terms are excellent at x = 0.5 and useless at x = 30.
FAQ
Why does accuracy fall off away from the expansion point?
Because the series is built from the function's behaviour at one point. Near that point a few terms capture it well; far away the truncation error grows, sometimes explosively.
What is the radius of convergence?
The distance from the expansion point within which the infinite series converges to the function. Outside it, the series diverges and adding terms makes things worse — the geometric series for 1/(1−x) beyond x = 1 is the standard example.
Where are Taylor series used in practice?
Numerical methods, physics approximations like small-angle sine, and inside the standard library functions your computer uses to evaluate exponentials and trigonometry.
How we compare
| Feature | Online Tool Store | A graphing calculator | A stats package |
|---|---|---|---|
| Interval of convergence | ✓ | ✗ | ✓ |
| Error at a chosen point | ✓ | ✗ | Sometimes |
| No CAS licence | ✓ | ✓ | ✗ |
| Symbolic manipulation | ✗ | ✓ | ✓ |
Taylor Series Expander reports the interval of convergence, because outside it every extra term makes the approximation worse.