What is a Taylor series?

A Taylor series expresses a smooth function as an infinite sum of terms derived from the function's derivatives at a single point a. Each term has the form f^(n)(a)/n! * (x-a)^n, where f^(n) is the n-th derivative.

What is a Maclaurin series?

A Maclaurin series is simply a Taylor series expanded around a = 0. It is the most common case and is what this tool computes by default when you leave the expansion point set to 0.

Why does ln(1+x) require x to be between -1 and 1?

The Taylor series for ln(1+x) has a radius of convergence of 1, meaning it only converges when |x| < 1 (and at x = 1 by conditional convergence). Outside that range the polynomial diverges from the true value.

How many terms should I use?

For most practical approximations, 5 to 10 terms give excellent accuracy near the expansion point. The tool supports up to 15 terms. The farther x is from a, the more terms you need.

Can I expand around a point other than 0?

Yes. Enter any value in the Expansion Point field. The tool recomputes the derivatives at a and builds the shifted polynomial (x - a)^n.

How is the approximation error calculated?

The tool evaluates your n-term polynomial at x and compares it with JavaScript's built-in Math.sin / Math.cos / Math.exp / Math.log, which use hardware-precision arithmetic. The difference is the truncation error.

Taylor Series Calculator

Select a function, choose an expansion point and number of terms, and instantly see the Taylor series polynomial with computed coefficients.

Parameters

Function

Expansion Point (a)

Set to 0 for a Maclaurin series.

Number of Terms (n): 6

115

Evaluate at x

Point where the approximation is evaluated.

Set the parameters and click Compute Series to see the expansion.

Polynomial Expression

Term Breakdown

n	Derivative f^(n)(a)	n!	Coefficient	Term

Numerical Approximation at x = 1

Polynomial

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Exact Value

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Abs. Error

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Summary

This Taylor Series Calculator computes the Taylor (or Maclaurin) series expansion for four common functions: sin(x), cos(x), e^x, and ln(1+x). You specify the expansion point a, the number of terms n, and an optional evaluation point x. The tool returns each term's coefficient and power, the full polynomial expression, and a numerical approximation you can compare against the exact value.

How it works

Select a function from the dropdown (sin, cos, exp, or ln(1+x)).
Enter the expansion point a (default 0 for a Maclaurin series).
Choose the number of terms n (1 to 15).
Optionally enter an evaluation point x to see the numerical approximation.
Click "Compute" — the tool displays each term, the full polynomial, and the approximation error.

Use cases

Verify hand-computed Taylor expansions for calculus homework.
Explore how adding more terms improves polynomial approximation accuracy.
Quickly generate Maclaurin series for sin, cos, and e^x.
Understand the radius of convergence by watching error grow far from the expansion point.
Prepare study notes for engineering math or physics courses.
Check approximations used in numerical methods and signal processing.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu