Taylor Series Calculator 2026
Instantly compute the polynomial expansion of any function around a center point. Includes step-by-step formulas, exact vs approximated values, and error analysis.
Function Specifications
Select a function and define the expansion parameters
The point around which the function is expanded (a=0 is Maclaurin).
Number of terms to compute (higher = more accurate).
The point at which to evaluate the polynomial.
Expansion Results
Polynomial formula, values, and error analysis
Select a function and click Calculate Taylor Series to see the polynomial expansion.
Common Maclaurin Series (a=0)
Standard Taylor series expansions centered at zero. These are fundamental in calculus for approximating functions.
| Function | Taylor Series Expansion | Radius of Convergence |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + ... | All real numbers |
| sin(x) | x - x³/3! + x⁵/5! - ... | All real numbers |
| cos(x) | 1 - x²/2! + x⁴/4! - ... | All real numbers |
| ln(1+x) | x - x²/2 + x³/3 - ... | -1 < x ≤ 1 |
| 1/(1-x) | 1 + x + x² + x³ + ... | |x| < 1 |
Taylor Series Calculator FAQ
Everything you need to know about polynomial expansions, convergence, and calculus approximations.
A Taylor series is a representation of a function as an infinite sum of terms that are expressed in terms of the derivatives of the function at a single point. It allows complex functions like sine, cosine, or exponentials to be approximated as polynomials, which are much easier to compute and analyse.
A Maclaurin series is simply a special case of a Taylor series where the center point (a) is zero. While a Taylor series can be expanded around any point 'a', a Maclaurin series is always expanded around a = 0. All Maclaurin series are Taylor series, but not all Taylor series are Maclaurin series.
The number of terms required depends on the function and how far your evaluation point (x) is from the center point (a). For points close to 'a', even 3 or 4 terms can provide extreme accuracy. For points further away, you may need 10, 20, or more terms. Functions like sine and cosine converge very quickly, while others may require many more terms.
The radius of convergence is the range of values for 'x' around the center point 'a' for which the Taylor series actually converges to the original function. For example, the series for ln(1+x) only converges if |x| < 1. If you evaluate the series outside this radius, the approximation will diverge and become completely inaccurate.
No, a function must be 'infinitely differentiable' at the center point to have a Taylor series. Furthermore, even if a function has a Taylor series, it might not converge to the original function everywhere. Functions with sharp corners, discontinuities, or vertical asymptotes at the center point cannot be expanded into a Taylor series.