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Taylor Series Calculator
Last updated: May 31, 2026
Written by Blake Boege
A Taylor series calculator is a mathematical tool that approximates a differentiable function with a polynomial expansion centered at a specific coordinate point. It calculates successive derivatives of the input function and evaluates them at the center to determine the coefficients using Taylor's theorem. The resulting polynomial provides a local approximation of the function's curve. Students, physicists, and engineers use this calculator to simplify complex functions, perform numerical approximations, solve differential equations, and evaluate complicated integrals in mathematical and physical models.
Approximate any differentiable function f(x) with a Taylor polynomial. Specify the center point, the number of terms, and view the successive derivatives step-by-step.
Quick Answer
Approximate a function f(x) centered around any point a with a Taylor polynomial. Enter the expression, center, and target terms to see derivatives and coefficients.
Derivative & Term Breakdown
| n | f⁽ⁿ⁾(x) | f⁽ⁿ⁾(a) | Term |
|---|---|---|---|
| 0 | e^x | 1 | 1 |
| 1 | e ^ x | 1 | x |
| 2 | e ^ x | 1 | x²/2 |
| 3 | e ^ x | 1 | x³/6 |
f(x) around x = 0 ≈
1 + x + x²/2 + x³/6
Approximated with 4 non-zero terms
Note: Taylor series are approximations. The accuracy increases as you add more terms or evaluate closer to the center point a.
Example: Taylor series of e^x around a=0 with 4 terms = 1 + x + x²/2 + x³/6
Examples
e^x around a=0, 4 terms
1 + x + x²/2 + x³/6
cos(x) around a=0, 3 terms
1 - x²/2 + x⁴/24
ln(x) around a=1, 3 terms
(x - 1) - (x - 1)²/2 + (x - 1)³/3
How it works
The Taylor polynomial of degree N-1 for a function f(x) centered at a is computed using the derivatives evaluated at a:
Taylor Polynomial Formula
f(x) ≈ Σ [f⁽ⁿ⁾(a) / n!] · (x - a)ⁿ
Term-by-term Expansion
f(a) + f′(a)(x-a) + [f″(a)/2!](x-a)² + [f‴(a)/3!](x-a)³ + ...
The Radius of Convergence
Not all Taylor series approximate a function over its entire domain. The interval of x-values for which the infinite Taylor series converges to the actual function is called the interval of convergence. For functions like $e^x$, $\sin(x)$, and $\cos(x)$, the series converges for all real numbers. For functions like $\ln(x)$ or $1/(1-x)$, the series only converges for a narrow range around the center point.
Practical Applications
Taylor series are used extensively in computational mathematics:
- Calculator Algorithms: Computers and calculators use Taylor series approximations to evaluate values of trigonometric and transcendental functions like $\sin(x)$ or $\ln(x)$ on a processor level.
- Differential Equations: Finding analytical solutions to differential equations can be difficult. Expressing variables as power series makes solving them much more manageable.
Related Calculus Calculators
For expansions specifically centered at zero, use our specialized Maclaurin Series Calculator. If you need to calculate individual derivatives, check out the Derivative Calculator or the multivariable Partial Derivative Calculator.
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Frequently asked questions
A Taylor series is an infinite sum of terms that represents a function. It is calculated from the values of the function's derivatives at a single point (the center).
This calculator calculates successive derivatives of your input function using symbolic differentiation. It then evaluates each derivative at the center point a to determine the coefficients of the polynomial.
A Maclaurin series is simply a special case of a Taylor series where the expansion point center is fixed at zero (a = 0).
The difference between the actual function value and the Taylor polynomial approximation is the remainder (or truncation error). The error is typically smaller for points closer to the center point a.
Taylor series allow engineers and scientists to approximate complex non-polynomial functions (like trigonometric or exponential functions) with simpler polynomial functions. This makes numerical analysis and integrations much easier.
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