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Power Series Calculator

Find the power series representation of functions centered at any point. Compute Taylor/Maclaurin coefficients, determine the radius and interval of convergence with endpoint analysis, and visualize how partial sums converge with an interactive animated graph.

Power Series Calculator
Examples:
POWER SERIES FORMULA
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$$
Choose a function to expand
Expansion point
0 to 20
Check approximation

Embed Power Series Calculator Widget

About Power Series Calculator

The Power Series Calculator finds the power series representation of mathematical functions centered at any point a. It computes the Taylor/Maclaurin expansion coefficients, determines the radius and interval of convergence (including endpoint analysis), displays a step-by-step derivation for each term, and provides an interactive animated graph showing how successive partial sums converge to the original function. This tool supports 11 common functions including exponential, trigonometric, logarithmic, and algebraic functions.

Key Concepts in Power Series

∑
Power Series
Σ aₙ(x−a)ⁿ — infinite polynomial
🎯
Center Point
Value a where the series is anchored
🔄
Radius R
Distance from center where series converges
↔
Interval
(a−R, a+R) with endpoint tests
📐
Coefficients
aₙ = f⁽ⁿ⁾(a) / n!
⚡
Convergence
Partial sums approach f(x)

Essential Formulas

ConceptFormulaDescription
Power Series\(f(x) = \sum_{n=0}^{\infty} a_n (x-a)^n\)General form centered at a
Taylor Coefficients\(a_n = \frac{f^{(n)}(a)}{n!}\)Coefficient from nth derivative
Radius of Convergence\(R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}\)Cauchy–Hadamard theorem
Ratio Test\(R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|\)Common method for finding R
Lagrange Remainder\(|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\)Error bound for partial sum

Understanding Power Series

A power series represents a function as an infinite sum of terms involving increasing powers of (x − a), where a is the center of expansion. The key idea is that if you know all the derivatives of a function at a single point a, you can reconstruct the entire function within the radius of convergence. Each coefficient aₙ = f⁽ⁿ⁾(a)/n! captures information about the function's curvature and higher-order behavior at the center. When a = 0, this is a Maclaurin series; for any other center, it is a Taylor series.

Radius and Interval of Convergence

Every power series has a radius of convergence R that determines where it converges. For |x − a| < R, the series converges absolutely; for |x − a| > R, it diverges. The radius equals the distance from the center a to the nearest singularity of the function in the complex plane. For example, 1/(1−x) centered at a = 0 has R = 1 because of the singularity at x = 1. The interval of convergence is (a − R, a + R), but the endpoints require separate testing using convergence tests like the alternating series test or p-series comparison.

How to Use the Power Series Calculator

  1. Select a function: Choose from the dropdown menu (e.g., eˣ, sin(x), ln(x), √x) or click a quick example button to auto-fill all fields.
  2. Enter the center point: Type the value of a. Use 0 for a Maclaurin series, or any other value like π, 1, or 4 for a general Taylor series.
  3. Set the number of terms: Enter n (0 to 20). More terms give better accuracy but produce longer expressions.
  4. Optionally evaluate: Enter an x value to compute the polynomial approximation P(x) and compare it to the actual function value f(x), with error analysis.
  5. Review results: Examine the polynomial expansion, interval of convergence (with number line visualization), coefficient table, step-by-step derivation, and interactive convergence graph. Use the slider or Animate button to watch partial sums progressively approximate the function.

Power Series vs. Taylor Series vs. Maclaurin Series

These terms describe related but distinct concepts. A power series is any series of the form Σ aₙ(x−a)ⁿ with arbitrary coefficients. A Taylor series is a power series whose coefficients come from the derivatives of a specific function: aₙ = f⁽ⁿ⁾(a)/n!. A Maclaurin series is a Taylor series with center a = 0. In practice, when people say "find the power series of f(x)," they usually mean the Taylor series. This calculator handles all three cases — set a = 0 for Maclaurin, any other value for a general Taylor expansion.

Applications of Power Series

Power series are fundamental tools in mathematics, physics, and engineering. They are used to approximate transcendental functions for numerical computation, solve differential equations (especially when closed-form solutions do not exist), evaluate limits and integrals of complex expressions, analyze the behavior of functions near specific points, and power modern scientific computing libraries. Many calculator chips internally use truncated power series to compute functions like sin, cos, exp, and log.

FAQ

What is a power series?
A power series is an infinite series of the form Σ aₙ(x−a)ⁿ, where a is the center, aₙ are the coefficients, and n ranges from 0 to infinity. It represents a function as an infinite polynomial. When the coefficients are derived from a function's derivatives (aₙ = f⁽ⁿ⁾(a)/n!), it becomes a Taylor series. When a = 0, it is called a Maclaurin series.
What is the interval of convergence?
The interval of convergence is the set of all x values for which the power series converges to a finite sum. It is always centered at the expansion point a and extends R units in each direction, where R is the radius of convergence. The endpoints x = a − R and x = a + R must be tested individually, as the series may converge at one, both, or neither endpoint.
How do you find the radius of convergence?
The radius of convergence R can be found using the Ratio Test (R = lim |aₙ/aₙ₊₁|) or the Root Test (R = 1/limsup |aₙ|^(1/n)). For Taylor series of known functions, R equals the distance from the center a to the nearest singularity in the complex plane. For example, ln(x) at a = 1 has R = 1 because x = 0 is a singularity one unit away.
What is the difference between a power series and a Taylor series?
A Taylor series is a specific type of power series where the coefficients are determined by the derivatives of a function: aₙ = f⁽ⁿ⁾(a)/n!. A general power series Σ aₙ(x−a)ⁿ can have any coefficients. In practice, the terms "power series" and "Taylor series" are often used interchangeably when the coefficients come from a known function's derivatives.
Why does a power series only converge in an interval?
A power series converges in a symmetric interval around its center because of singularities — points where the function is undefined or not analytic — in the complex plane. The radius of convergence equals the distance from the center to the nearest such singularity. Even if the function is well-behaved on the real line, a complex singularity at the same distance can limit convergence. For instance, 1/(1+x²) has no real singularities, but has complex singularities at x = ±i, giving R = 1 when centered at 0.

Reference this content, page, or tool as:

"Power Series Calculator" at https://MiniWebtool.com/power-series-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

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