Taylor Series Calculator — Expand Functions into Power Series Instantly

Generate Taylor and Maclaurin series expansions for common functions around any center point. Free online taylor series calculator with step-by-step derivative evaluation and formula breakdown.

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Taylor Series Calculator

Select a function, choose a center point, and specify the number of terms to generate the Taylor series expansion.

Select a function and click Calculate Taylor Series to see the expansion.

Taylor Series Formula Explained

The Taylor series of a function f(x) around a center point a is an infinite sum that represents the function as a power series. Each term uses the function's derivatives evaluated at a.

f(x) = Σ f⁽ⁿ⁾(a) · (x − a)ⁿ / n!   for n = 0, 1, 2, …
Maclaurin Series (a=0):   f(x) = Σ f⁽ⁿ⁾(0) · xⁿ / n!

Variable Definitions

  • a — The center point around which the series is expanded
  • f⁽ⁿ⁾(a) — The n-th derivative of f evaluated at x = a
  • n! — n factorial (n! = 1 × 2 × … × n), with 0! = 1
  • (x − a)ⁿ — The n-th power of the distance from the center
  • Σ — Summation over all terms from n = 0 to ∞

The Taylor polynomial (truncated to a finite number of terms) provides a polynomial approximation that matches the function's value and derivatives at the center point.

How to Calculate a Taylor Series Expansion

Follow these steps to compute a Taylor series for any differentiable function:

  1. Choose the center point a — Use a=0 for a Maclaurin series (simplest form).
  2. Compute successive derivatives — Find f(a), f'(a), f''(a), f'''(a), and so on.
  3. Divide by the factorial — The coefficient for term n is f⁽ⁿ⁾(a) / n!.
  4. Multiply by (x−a)ⁿ — Each term includes the appropriate power of (x−a).
  5. Sum all terms — Add terms from n=0 up to the desired degree of approximation.

For example, the Maclaurin series for eˣ has f⁽ⁿ⁾(0)=1 for all n, giving: eˣ ≈ 1 + x + x²/2! + x³/3! + x⁴/4! + …

Taylor Series Calculator Examples

Example 1: Maclaurin Series for sin(x)

Expand sin(x) around a=0 with 5 terms.

sin(0)=0, sin'(0)=1, sin''(0)=0, sin'''(0)=−1, sin''''(0)=0
sin(x) ≈ x − x³/3! + x⁵/5! = x − x³/6 + x⁵/120

Example 2: Taylor Series for eˣ at a=1

Expand eˣ around a=1 with 4 terms.

All derivatives of eˣ equal eˣ, so f⁽ⁿ⁾(1)=e
eˣ ≈ e + e(x−1) + e(x−1)²/2! + e(x−1)³/3!

Example 3: Maclaurin Series for 1/(1−x)

Expand 1/(1−x) around a=0 with 5 terms (geometric series).

f⁽ⁿ⁾(0)=n! for all n, so coefficient = 1
1/(1−x) ≈ 1 + x + x² + x³ + x⁴   (|x|<1)

Real-World Taylor Series Applications

  • Physics & Engineering: Approximating complex functions like pendulum motion (sinθ ≈ θ for small angles) in classical mechanics.
  • Numerical Analysis: Deriving finite difference formulas and estimating truncation errors in computational methods.
  • Signal Processing: Linearizing nonlinear systems around operating points for filter design and control systems.
  • Economics: Approximating utility and production functions with quadratic Taylor expansions for optimization.
  • Machine Learning: Gradient-based optimization and loss function approximations using second-order Taylor expansions (Newton's method).
  • Computer Graphics: Fast polynomial approximations for trigonometric functions in shader programming.
  • Limit Evaluation: Calculating limits using Taylor series expansions to resolve indeterminate forms.

People Also Ask

A Taylor series is an infinite sum of terms that expresses a function as a power series around a specific center point a. Each term involves the function's derivatives evaluated at a, divided by factorials, multiplied by powers of (x−a). The Taylor polynomial provides increasingly accurate polynomial approximations of the original function near the center point.
A Maclaurin series is a special case of the Taylor series where the center point a=0. So every Maclaurin series is a Taylor series, but Taylor series can be centered at any point a. The general formula is f(x)=Σf⁽ⁿ⁾(a)(x−a)ⁿ/n!, and setting a=0 gives the Maclaurin series: f(x)=Σf⁽ⁿ⁾(0)xⁿ/n!.
The number of terms needed depends on the function, distance from the center point, and desired accuracy. For values close to the center, 3–5 terms often suffice. For values farther from the center or rapidly changing functions, 8–12 or more terms may be required. The Lagrange remainder formula estimates the truncation error.
The radius of convergence is the distance from the center point a within which the Taylor series converges to the function. For 1/(1−x) at a=0, the radius is 1 (converges for |x|<1). For sin(x) and eˣ, the radius is infinite—the series converges for all real x.
Yes, for analytic functions like sin(x), cos(x), and eˣ, the infinite Taylor series converges exactly to the function for all x within the radius of convergence. When truncated to a finite polynomial, we get an approximation whose accuracy improves as more terms are added.

Frequently Asked Questions

Yes. Simply set the center point a=0, and the calculator generates the Maclaurin series expansion. This is the most common use case and works for all supported functions.
The calculator supports sin(x), cos(x), eˣ, ln(1+x), 1/(1−x), arctan(x), sinh(x), and cosh(x). Each function uses exact derivative formulas evaluated at your chosen center point for precise coefficient computation.
The remainder (or error) term Rₙ(x) measures the difference between the true function value and the truncated Taylor polynomial. The Lagrange form states Rₙ(x)=f⁽ⁿ⁺¹⁾(c)(x−a)ⁿ⁺¹/(n+1)! for some c between a and x, giving a bound on the approximation error.
The radius of convergence R can be found using the ratio test: R=1/limsup|cₙ|¹ᐟⁿ, where cₙ are the Taylor coefficients. It tells you the interval (a−R, a+R) where the series is guaranteed to converge to the function.
Absolutely. The calculator accepts any real number as the center point, including negative values and decimals. Just ensure the function is defined at that point (e.g., ln(1+x) requires a>−1, and 1/(1−x) requires a≠1).
A second-order Taylor approximation includes terms up to n=2: f(x)≈f(a)+f'(a)(x−a)+f''(a)(x−a)²/2. This quadratic approximation captures curvature and is widely used in optimization (Newton's method) and physics (harmonic oscillator approximation).

Taylor Series Glossary

Taylor Series

An infinite power series representation of a function around a center point a, using the function's derivatives at a.

Maclaurin Series

A Taylor series centered at a=0. The simplest and most common form of Taylor expansion.

Radius of Convergence

The distance from the center within which the Taylor series converges to the function. Determined by the ratio test.

Factorial (n!)

The product of all positive integers up to n: n!=1×2×…×n. Appears in the denominator of each Taylor term.

Analytic Function

A function that equals its Taylor series within the radius of convergence. sin(x), cos(x), and eˣ are analytic everywhere.

Lagrange Remainder

A formula for the error when truncating a Taylor series: Rₙ(x)=f⁽ⁿ⁺¹⁾(c)(x−a)ⁿ⁺¹/(n+1)! for some c between a and x.

Taylor Polynomial

The finite sum of the first N+1 terms of a Taylor series. Used to approximate functions with polynomial expressions.

Power Series

An infinite series of the form Σcₙ(x−a)ⁿ. Taylor series are a specific type of power series where coefficients involve derivatives.

Editorial Review & Methodology

This Taylor series calculator was built and reviewed by the NumbrWiz Editorial Team. The Taylor series expansion is a cornerstone of calculus and numerical analysis, verified against standard curricula including AP Calculus BC, college-level real analysis, and advanced engineering mathematics textbooks.

  • Derivative verification: All derivative formulas are cross-checked against authoritative calculus references and symbolic computation results.
  • Coefficient accuracy: Each Taylor coefficient is computed using exact derivative evaluation at the specified center point with proper factorial normalization.
  • Domain validation: The calculator checks for valid center points within each function's domain before computing the series.

Transparency note: All calculations run client-side in your browser. No data is collected, stored, or transmitted. Results are for educational purposes; for mission-critical applications, verify expansions with analytical methods or symbolic algebra systems.

Page last reviewed: May 2026 · NumbrWiz Editorial Team