Maclaurin Series Calculator

Compute the Maclaurin series expansion of common functions at x = 0. Get the nth-order polynomial terms, a Lagrange remainder estimate, the radius of convergence and an animated graph of partial sums converging to the function.

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About Maclaurin Series Calculator

The Maclaurin Series Calculator computes the Maclaurin series expansion of common mathematical functions centered at x = 0. It generates the nth-order polynomial approximation, displays a complete coefficient table, provides Lagrange remainder estimates for error analysis, shows the radius of convergence, and features an interactive animated graph that visualizes how partial sums progressively converge to the original function.

Common Maclaurin Series Expansions

ℯ

eˣ

1 + x + x²/2! + x³/3! + ...

∿

sin(x)

x − x³/3! + x⁵/5! − ...

∾

cos(x)

1 − x²/2! + x⁴/4! − ...

㏑

ln(1+x)

x − x²/2 + x³/3 − ...

∡

arctan(x)

x − x³/3 + x⁵/5 − ...

∞

1/(1−x)

1 + x + x² + x³ + ...

Key Formulas

Concept	Formula	Description
Maclaurin Series	\(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\)	Taylor series at a = 0
nth Coefficient	\(a_n = \frac{f^{(n)}(0)}{n!}\)	Coefficient of xⁿ
Lagrange Remainder	\(\|R_n(x)\| \leq \frac{M \|x\|^{n+1}}{(n+1)!}\)	Upper bound on truncation error
Radius of Convergence	\(R = \frac{1}{\limsup_{n \to \infty} \|a_n\|^{1/n}}\)	Range where series converges

Understanding Maclaurin Series

A Maclaurin series represents a function as an infinite polynomial by using information about the function's derivatives at x = 0. The zeroth term is simply f(0), the first-order term captures the slope f'(0), the second-order term captures the curvature f''(0)/2!, and so on. Each additional term refines the approximation, matching one more derivative at the origin. Within the radius of convergence, the infinite sum equals the function exactly.

How to Use the Maclaurin Series Calculator

Select a function: Choose from the dropdown (e.g., sin(x), eˣ, ln(1+x)) or click a quick example button to auto-fill the form.
Enter the number of terms: Specify n (0 to 20) for the polynomial order. Higher n gives better accuracy but more terms.
Optionally enter an x value: Type a number to evaluate the polynomial and compare it against the exact function value, with error analysis.
Click Expand Series: Press the button to compute the Maclaurin expansion instantly.
Explore the results: Review the polynomial formula, coefficient table, and step-by-step derivation. Use the slider or Animate button on the convergence graph to watch how adding terms progressively approximates the function.

Maclaurin vs. Taylor Series

The Taylor series generalizes polynomial approximation to any center point a: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\). The Maclaurin series is the special case where a = 0, simplifying the formula to \(f(x) = \sum \frac{f^{(n)}(0)}{n!} x^n\). While a Taylor series can be centered anywhere to improve convergence near a specific point, the Maclaurin series is often preferred for functions with simple derivatives at zero, such as sin(x), cos(x), and eˣ.

Convergence and the Radius of Convergence

Every power series has a radius of convergence R. For |x| < R the series converges absolutely; for |x| > R it diverges. Some series (like eˣ, sin(x), cos(x)) converge for all real x, so R = ∞. Others (like ln(1+x), 1/(1−x), arctan(x)) have R = 1, meaning they only converge within the interval (−1, 1) or [−1, 1]. The interactive graph shows convergence radius boundaries as red dashed lines.

Lagrange Remainder and Error Bounds

The Lagrange remainder \(R_n(x)\) quantifies the truncation error when using the first n+1 terms. Its bound is \(|R_n(x)| \leq \frac{M |x|^{n+1}}{(n+1)!}\), where M is the maximum of \(|f^{(n+1)}(t)|\) on the interval [0, x]. For functions like eˣ and sin(x), where all derivatives are bounded, this provides a tight guarantee on accuracy. The factorial growth in the denominator means the error decreases rapidly as n increases.

FAQ

What is a Maclaurin series?

A Maclaurin series is a Taylor series expansion of a function centered at x = 0. It represents a function as an infinite sum of terms: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... Each term involves a higher-order derivative of the function evaluated at zero, divided by the corresponding factorial.

What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is a special case of a Taylor series where the expansion point is a = 0. A Taylor series can be centered at any point a, using the formula f(x) = Σ f⁽ⁿ⁾(a)(x−a)ⁿ/n!. When a = 0, the Taylor series becomes a Maclaurin series. The Maclaurin series is simpler because the (x−a)ⁿ terms reduce to just xⁿ.

What is the radius of convergence?

The radius of convergence R is the distance from the center (x = 0 for Maclaurin series) within which the series converges to the actual function value. For |x| < R, the series converges; for |x| > R, it diverges. Some series like eˣ, sin(x), and cos(x) converge for all real numbers (R = ∞), while others like ln(1+x) and 1/(1−x) have R = 1.

How is the Lagrange remainder used?

The Lagrange remainder R_n(x) gives an upper bound on the error when approximating f(x) with the first n+1 terms. The formula is |R_n(x)| ≤ M × |x|^(n+1) / (n+1)!, where M is the maximum of |f^(n+1)(t)| for t between 0 and x. A smaller remainder means a better approximation. This is essential in numerical analysis for guaranteeing precision.

Why do some Maclaurin series only have odd or even powers?

This happens because of the function's symmetry. Odd functions like sin(x) and arctan(x) satisfy f(−x) = −f(x), so their even-order derivatives at 0 are all zero, leaving only odd powers. Even functions like cos(x) and cosh(x) satisfy f(−x) = f(x), so their odd-order derivatives at 0 are zero, leaving only even powers. This pattern is visible in the coefficient table.

Reference this content, page, or tool as:

"Maclaurin Series Calculator" at https://MiniWebtool.com/maclaurin-series-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

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