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Taylor Series Calculator

Expand a function into a Taylor series around center a up to term n

Select Function

e^x sin(x) cos(x) ln(1+x)

Center a = , Terms (0 to n): n =

Taylor Series Formula

f(x) = sum_{n=0}^{inf} f^(n)(a) * (x-a)^n / n!

Term 0: f(a) (constant)

Term 1: f(a)*(x-a) (linear)

Term 2: f(a)*(x-a)^2/2 (quadratic)

The Taylor series approximates a function using its derivatives at a single point. Adding more terms improves the approximation near the center point. The series is useful for numerical computation of transcendental functions.

⚠ The Taylor series converges to f(x) only within its radius of convergence. Outside that radius, the series may diverge.

What Is a Taylor Series?

A Taylor series expands a function into an infinite polynomial around a center point. Each term involves the nth derivative evaluated at the center, divided by n factorial. The series provides increasingly accurate approximations as more terms are added.

Definition

f(x)=sum f^(n)(a)(x-a)^n/n!. Requires all derivatives at a. Generalizes linear approximation to higher orders.

Convergence

e^x, sin, cos: converge for all x. 1/(1-x): converges for |x|<1. ln(1+x): converges for |x|<=1, x!=-1.

Error Term

The remainder R_n(x) = f^(n+1)(c)*(x-a)^(n+1)/(n+1)! for some c between a and x. Bounding this gives approximation accuracy.

Applications

Numerical computation of functions (calculators use Taylor), solving differential equations, physics approximations (pendulum, relativity).

Teaching Example: e^x at a=0, n=4. Terms: f(0)=1 (n=0), f(0)=1 (n=1), f(0)/2!=1/2 (n=2), f(0)/3!=1/6 (n=3), f^(4)(0)/4!=1/24 (n=4). Series: 1 + x + x^2/2 + x^3/6 + x^4/24.

Applications

Numerical Methods Physics Engineering Calculus Computer Science

Frequently Asked Questions

What is Taylor series?▼

Function expansion using derivatives at a point: sum f^(n)(a)*(x-a)^n/n!. Approximates f near a.

Taylor vs Maclaurin?▼

Maclaurin is Taylor with center a=0. Taylor can be centered anywhere. Maclaurin is the special case.

Radius of convergence?▼

Distance from center where series converges. e^x: inf. 1/(1-x): 1. Depends on function singularities.

Why use Taylor series?▼

Computers calculate e^x, sin(x) using Taylor series. Also used in approximations, limits, integrals, and differential equations.

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