maclaurin expansion calculator

Generate a Maclaurin polynomial P_n(x) for a common function and (optionally) evaluate the approximation at a specific value of x.

Choose function f(x)

Order n (0 to 20)

Evaluate at x (optional)

What is a Maclaurin expansion?

A Maclaurin expansion is a special case of a Taylor series centered at x = 0. It represents a function as an infinite polynomial:

f(x) = f(0) + f'(0)x + f''(0)x²/2! + ...

In practice, we usually stop after a finite number of terms (order n). That finite polynomial is called the Maclaurin approximation, and it often gives highly accurate values near zero.

How to use this calculator

Select a function (such as e^x, sin(x), or ln(1+x)).
Pick the expansion order n.
Optionally enter a value of x to compare approximation vs exact value.
Click Calculate Expansion.

The calculator returns:

The polynomial P_n(x)
Non-zero coefficients used in the expansion
Approximate value at your chosen x
Exact value and absolute error when available

Common Maclaurin series included

1) Exponential

e^x = 1 + x + x^2/2! + x^3/3! + ...

2) Sine

sin(x) = x - x^3/3! + x^5/5! - ...

3) Cosine

cos(x) = 1 - x^2/2! + x^4/4! - ...

4) Natural log

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...

5) Geometric form

1/(1-x) = 1 + x + x^2 + x^3 + ...

6) Arctangent

arctan(x) = x - x^3/3 + x^5/5 - ...

Why order n matters

A larger order usually means better accuracy near x=0 because you keep more terms from the infinite series. However, bigger n also means more computation and may not help if your input is outside the convergence interval.

Near zero: low-order polynomials are often excellent.
Far from zero: you may need a higher order (or a different method).
Outside convergence radius: the series may fail to converge.

Applications of Maclaurin approximations

Maclaurin and Taylor approximations are used throughout mathematics, physics, economics, machine learning, signal processing, and engineering:

Approximating difficult functions quickly
Analyzing local behavior and sensitivity
Deriving numerical methods and algorithms
Simplifying differential equations in modeling
Building error bounds and stability checks

Quick accuracy tip

If your approximation error is too large, try one (or both) of these:

Increase the order n
Evaluate at values of x closer to 0

For many practical tasks, even a modest order (like 4 to 8) can be remarkably effective.