Taylor Series Calculator

Compute Taylor series coefficients and polynomial approximations for any function. Shows Lagrange remainder bound, radius of convergence hint, and convergence of partial sums.

Taylor polynomial Pₙ(x) at given x (numerical approx.)
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True f(x) at given x —
Error |f(x) − Pₙ(x)| —
Coefficients c₀, c₁, c₂, ... (f^(k)(a)/k!) —
Extended More scenarios, charts & detailed breakdown ▾
Taylor polynomial at x (numerical approx.)
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Coefficients c₀…cₙ —
Professional Full parameters & maximum detail ▾

Polynomial Approximation

Taylor polynomial at x (numerical approx.) —
True f(x) —
Actual error |f − Pₙ| —

Remainder & Convergence

Lagrange remainder bound M·|x-a|^(n+1)/(n+1)! —
Partial sums n=2,4,6,8 —

How to Use This Calculator

  1. Enter f(x) (e.g. exp(x), sin(x)).
  2. Set the expansion point a (default 0 for Maclaurin).
  3. Set the number of terms n (1–10).
  4. Enter an x value to evaluate Pₙ(x) and compare with f(x).
  5. The Professional tab shows Lagrange remainder bound and partial sum convergence.

Formula

Taylor series: f(x) = Σₖ₌₀^∞ f⁽ᵏ⁾(a)/k! · (x−a)ᵏ

Lagrange remainder: |Rₙ| ≤ M|x−a|ⁿ⁺¹/(n+1)!

Example

eˣ at a=0, n=5, x=1: P₅(1) = 1+1+1/2+1/6+1/24+1/120 = 2.716667. True e ≈ 2.71828. Error ≈ 0.00162.

Frequently Asked Questions

  • A Taylor series expands a function f(x) around a point a as an infinite sum: Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. The partial sum Pₙ(x) is the Taylor polynomial of degree n.
  • The Lagrange form of the remainder Rₙ(x) = f⁽ⁿ⁺¹⁾(c)/(n+1)! · (x−a)ⁿ⁺¹ for some c between a and x. It bounds the error |f(x) − Pₙ(x)| ≤ M|x−a|ⁿ⁺¹/(n+1)! where M bounds |f⁽ⁿ⁺¹⁾|.
  • The Taylor series converges only within some radius R around a. For eˣ, sin, cos the radius is ∞. For ln(1+x) it is R=1 (converges for |x|<1). The ratio test on consecutive terms gives R.
  • The k-th coefficient cₖ = f⁽ᵏ⁾(a)/k!, computed by repeated numerical differentiation using the central difference formula.
  • A Maclaurin series is a Taylor series expanded at a=0. The Maclaurin Series Calculator focuses on the special case a=0 and provides exact coefficient formulas for common functions.

Related Calculators

Σ Maclaurin Series Calculator 📐 Derivative Calculator → Limit Calculator
Sources & References (5) ▾
  1. Taylor Series — Paul's Online Math Notes — Lamar University
  2. MIT OCW 18.01 Taylor Series — MIT
  3. NIST DLMF Chapter 1 — Algebraic and Analytic Methods — NIST
  4. Taylor & Maclaurin Series — Khan Academy — Khan Academy
  5. Wolfram MathWorld — Taylor Series — Wolfram MathWorld