Maclaurin Series Calculator

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

Key Formulas

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

1. 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.
2. Enter the number of terms: Specify n (0 to 20) for the polynomial order. Higher n gives better accuracy but more terms.
3. Optionally enter an x value: Type a number to evaluate the polynomial and compare it against the exact function value, with error analysis.
4. Click Expand Series: Press the button to compute the Maclaurin expansion instantly.
5. 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.

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