Taylor Series
≈
Tap a term to see what it means, or .
Calculate
Example: approximate e = e¹ by adding only the terms up to x⁴/4!. How close does it get? Change the function, the point and the degree to work out your own case.
Step by step
What each term means
- Function (f(x))
- The function to approximate. Here the series is around 0 (also called the Maclaurin series), so the approximation is best near x = 0.
- Sum up to Degree N
- Adds the terms from degree 0 to N. Each extra term corrects the previous ones; with infinitely many terms, the sum reproduces the function wherever the series converges.
- Derivative at 0 (f⁽ᵏ⁾(0))
- The k-th derivative of the function at 0. It makes the polynomial have, at 0, the same value, the same slope, the same curvature and so on as the function.
- Factorial (k!)
- Makes up for the fact that differentiating xᵏ k times gives k!. That is why the coefficients shrink fast and high-degree terms weigh little near 0.
- Power (xᵏ)
- Carries the term away from 0. The farther the point, the larger the powers and the more terms are needed for a good approximation.