Taylor Series Calculator — Step-by-Step Solutions

f(x)

Centre (a)

Number of Terms (1-15)

Your calculations will appear here

The Taylor Series Calculator computes the Taylor series expansion of a function around a specified point. A Maclaurin series is the special case where the expansion point is zero.

Enter your function, the variable, the expansion point, and the number of terms. The calculator computes successive derivatives to build the polynomial approximation.

How to Use the Taylor Series Calculator

Enter the function to expand (e.g. sin(x), e^x, ln(1+x)).
Specify the expansion point (0 for Maclaurin series).
Choose the number of terms (up to 15).

Formula and Theory

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. When the expansion point is 0, it is called a Maclaurin series.

The nth term of a Taylor series around point a is: f^(n)(a) / n! * (x - a)^n. The more terms included, the better the approximation near the expansion point.

Worked Examples

Example 1: Exponential function

Problem: Find the Maclaurin series of e^x (first 5 terms)

Solution: 1 + x + x^2/2! + x^3/3! + x^4/4!

Answer: 1 + x + x^2/2 + x^3/6 + x^4/24

Example 2: Sine function

Problem: Find the Maclaurin series of sin(x) (first 4 non-zero terms)

Solution: x - x^3/3! + x^5/5! - x^7/7!

Answer: x - x^3/6 + x^5/120 - x^7/5040

Frequently Asked Questions

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a special case of a Taylor series where the expansion point is 0. A Taylor series can be expanded around any point a.

What is the radius of convergence?

The radius of convergence is the distance from the expansion point within which the Taylor series converges to the function. Outside this radius, the series may diverge.