Taylor Series Calculator

How to Use the Taylor Series Calculator

Select the function you want to expand from the dropdown menu, then choose how many terms to include in the expansion (1 to 20). The calculator instantly displays the polynomial approximation along with each individual term and its coefficient. The expansion is centred at x = 0 (Maclaurin series).

For example, selecting sin(x) with 4 terms produces: x − x³/6 + x⁵/120 − x⁷/5040 + ... This corresponds to the well-known power series for sine.

Understanding Taylor Series

The Taylor series of a function f(x) about a point a is the infinite sum: f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + f′′′(a)(x−a)³/3! + ... Each term involves successively higher derivatives of f evaluated at a. This provides a polynomial approximation that matches the function and all its derivatives at the centre point.

Convergence

Not all Taylor series converge everywhere. The radius of convergence determines the interval where the series gives a valid approximation. For e^x, sin(x), and cos(x), the series converges for all real numbers. For ln(1+x), convergence is limited to −1 < x ≤ 1. For 1/(1−x), convergence requires |x| < 1.

Applications

Taylor series are used extensively in numerical methods (approximating functions by polynomials), physics (perturbation theory, small-angle approximations), engineering (linearisation around operating points), and computer science (efficient function evaluation in hardware). The series for e^x, sin, and cos are computed by CPUs and math libraries to evaluate these functions efficiently.

Frequently Asked Questions

What is a Taylor series?

An infinite polynomial expansion of a function centred at a point, using successively higher derivatives to match the function more closely.

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a Taylor series centred at a = 0. This calculator computes Maclaurin series for supported functions.

Which functions are supported?

e^x, sin(x), cos(x), ln(1+x), 1/(1−x), sinh(x), cosh(x), and atan(x).

How many terms should I use?

For most purposes, 5 to 10 terms give excellent accuracy near the centre. More terms extend the accurate range.

What is the radius of convergence?

e^x, sin, cos, sinh, cosh converge for all x. ln(1+x) converges for −1 < x ≤ 1. 1/(1−x) converges for |x| < 1. atan(x) converges for |x| ≤ 1.