What is a Maclaurin series?

A Maclaurin series is a special case of the Taylor series where the expansion point is x = 0. It expresses a function as an infinite sum of terms based on the function's derivatives at zero: f(x) = f(0) + f'(0)x + f''(0)x²/2! + ….

Why does the ln(1+x) series have a limited convergence range?

The series for ln(1+x) converges only for -1 1 the approximation error will grow, not shrink, as you add more terms.

How many terms do I need for a good approximation?

It depends on the function and the value of x. Near x = 0 even 4–5 terms give excellent accuracy. Farther from zero you need more terms. The "Error" row in the output shows the absolute difference between the partial sum and the exact value.

Can I use this for tan(x)?

Yes. tan(x) is included. Its Maclaurin series has a more complex pattern of coefficients (Bernoulli numbers) and converges only for |x| < π/2 ≈ 1.5708. For x near ±π/2 the partial sum diverges.

What does "exact value" mean in the output?

The exact value is computed using the browser's built-in Math functions (Math.sin, Math.exp, etc.), which use IEEE 754 double precision — accurate to about 15 significant digits. It is not a symbolic exact result.

Maclaurin Series Calculator

Name: Maclaurin Series Calculator
Availability: InStock
Author: Nham Vu

Choose a function, set the number of terms, and see the Maclaurin series expansion with each term listed and evaluated at any x.

Series Settings

Function

Number of terms: 6

Evaluate at x =

Term Breakdown

n	Term (symbolic)	Coefficient	Value at x	Running sum

Select a function and click Calculate

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Summary

Choose a function, set the number of terms, and see the Maclaurin series expansion with each term listed and evaluated at any x.

How it works

Select a function from the dropdown (sin, cos, e^x, ln(1+x), etc.).
Choose how many terms to include in the expansion (2–20).
Optionally enter an x value to evaluate the partial sum numerically.
The calculator displays the series formula, each term with its coefficient and power, and the computed partial sum vs. the exact value.

Use cases

Check series expansions for calculus homework or exams.
Visualize how quickly a Maclaurin series converges to the exact function.
Derive polynomial approximations for engineering calculations.
Explore truncation error between partial sums and exact values.
Understand the pattern of coefficients in common series expansions.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu