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Calculator Answers

Maclaurin Series Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A Maclaurin series calculator is a calculus utility that computes the polynomial expansion of an infinitely differentiable function centered specifically at zero. It is a special case of the Taylor series where the expansion point is fixed. The tool iteratively calculates the function's derivatives and evaluates them at zero to compile the terms. It is widely used in trigonometry, exponential equations, computer science, and engineering to evaluate transcendental functions near zero using simple polynomial arithmetic.

Approximate any differentiable function f(x) with a Maclaurin series. See the step-by-step table of derivatives evaluated at zero.

See also: Taylor Series Calculator, Infinite Series Calculator, and Series Calculator.

Quick Answer

Calculate the Maclaurin series expansion of a function f(x) centered at zero. Enter the function and the number of terms to see the polynomial.

Derivative & Term Breakdown (at x=0)

nf⁽ⁿ⁾(x)f⁽ⁿ⁾(0)Term
1cos(x)1x
3-cos(x)-1-x³/6
5cos(x)1x⁵/120
7-cos(x)-1-x⁷/5040
Maclaurin Series Approximation

f(x) ≈

x - x³/6 + x⁵/120 - x⁷/5040

Approximated with 4 non-zero terms

Function f(x)sin(x)
Center Point (a)0
Target Terms4
Actual Terms Found4

Note: A Maclaurin series is a Taylor series centered at zero (a = 0).

Example: Maclaurin series of sin(x) with 4 terms = x - x³/6 + x⁵/120 - x⁷/5040

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Examples

sin(x), 4 terms

x - x³/6 + x⁵/120 - x⁷/5040

cos(x), 4 terms

1 - x²/2 + x⁴/24 - x⁶/720

e^x, 4 terms

1 + x + x²/2 + x³/6

1/(1 - x), 4 terms

1 + x + x² + x³

How it works

The Maclaurin series is calculated by evaluating the function and its derivatives at x = 0:

Maclaurin Series Formula

f(x) ≈ Σ [f⁽ⁿ⁾(0) / n!] · xⁿ

Expansion

f(0) + f′(0)x + [f″(0)/2!]x² + [f‴(0)/3!]x³ + ...

Why Expand at Zero?

Evaluating derivatives at zero is algebraically much simpler than evaluating them at other values. Many of the most common power series used in physics and calculus, such as those for ex, sin(x), and cos(x), are Maclaurin series. They provide highly accurate local approximations for small values of x (values close to zero).

Related Mathematical Estimators

For expansions around center values other than zero, use the general Taylor Series Calculator. For basic rates of change, visit our Rate of Change Calculator.

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Frequently asked questions

A Maclaurin series is a Taylor series expansion of a mathematical function about 0. It represents a function as an infinite sum of terms calculated from its derivatives at zero.

This calculator takes your function, calculates symbolic derivatives using computer algebra math rules, and evaluates them at x = 0 to build the polynomial approximation.

It is named after Colin Maclaurin, a Scottish mathematician who used this special case of the Taylor series extensively in the 18th century, though it was originally discovered by Brook Taylor.

No, a function must be infinitely differentiable at x = 0 to have a Maclaurin series. For example, ln(x) cannot be expanded as a Maclaurin series because ln(0) and its derivatives are undefined.