Power Series Calculator
Analyze power series convergence, find radius of convergence, and generate Taylor series representations with comprehensive analysis. Our calculus calculator supports convergence testing, interval determination, series representation, and advanced mathematical analysis for power series.
Last updated: December 15, 2024
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Enter coefficients separated by commas. Use fractions like 1/2!, 1/3! for factorials
Center point of the power series
Variable name for the series
Power Series Form:
Σ(n=0 to ∞) aₙ(x)ⁿ
Series Analysis
Series Representation:
1 + x + x²/2! + x³/3! + ...
Analysis Type:
Convergence analysis
Radius of Convergence:
R = ∞
Method: Ratio Test
Interval of Convergence:
(-∞, ∞)
Partial Sums:
Analysis:
The power series has infinite radius of convergence (R = ∞), meaning it converges for all real values of x. This is characteristic of entire functions like the exponential series.
Power Series Tips:
- • Coefficients: Use fractions like 1/2!, 1/3! for factorial denominators
- • Convergence: Ratio test: R = 1/lim|aₙ₊₁/aₙ|, Root test: R = 1/lim|aₙ|^(1/n)
- • Taylor Series: aₙ = f⁽ⁿ⁾(a)/n! where f⁽ⁿ⁾ is the nth derivative
- • Interval: Series converges absolutely for |x-a| < R
Quick Example Result
For the exponential power series Σ(n=0 to ∞) xⁿ/n!:
R = ∞
Converges for all x ∈ (-∞, ∞) using Ratio Test
How This Calculator Works
Our power series calculator analyzes infinite series of the form Σ(n=0 to ∞) aₙ(x-a)ⁿ using advanced convergence tests and mathematical algorithms. The calculator applies the ratio test, root test, and Cauchy-Hadamard theorem to determine convergence properties, generate series representations, and provide comprehensive mathematical analysis.
Power Series Analysis Algorithm
Extract and validate coefficient sequence aₙUse Ratio Test or Root Test to find radius RFind interval of convergence and test endpointsCreate series representation and partial sumsThe power series analysis employs sophisticated mathematical algorithms to determine convergence properties. The Ratio Test computes R = 1/lim|aₙ₊₁/aₙ|, while the Root Test uses R = 1/lim sup|aₙ|^(1/n). For Taylor series, coefficients follow aₙ = f⁽ⁿ⁾(a)/n! where f⁽ⁿ⁾ represents the nth derivative at the center point a.
Mathematical foundation for power series convergence analysis
Mathematical Foundation
Power series form the foundation of analytic function theory and provide powerful tools for function approximation, differential equation solutions, and mathematical analysis. The convergence behavior is governed by the Cauchy-Hadamard theorem, which establishes the radius of convergence as the reciprocal of the limit superior of the nth root of the absolute values of coefficients. Within the radius of convergence, power series can be differentiated and integrated term by term, making them invaluable for calculus operations.
- Convergence tests determine the domain where the series represents a function
- Taylor series provide polynomial approximations of smooth functions
- Power series enable analytical solutions to differential equations
- Series operations preserve convergence properties within appropriate domains
Sources & References
- Real and Complex Analysis - Walter RudinComprehensive treatment of power series convergence theory and applications
- Wolfram MathWorld - Power Series ReferenceDetailed mathematical reference for power series properties and convergence tests
- Advanced Calculus - Patrick M. FitzpatrickAdvanced treatment of infinite series and convergence analysis in calculus
Need help with other series analysis? Check out our Taylor series calculator and convergence calculator.
Get Custom Calculator for Your PlatformExample Analysis
Series Definition:
- Function: f(x) = e^x
- Center: a = 0 (Maclaurin series)
- Coefficients: aₙ = 1/n!
- Series: Σ(n=0 to ∞) xⁿ/n!
Convergence Analysis:
Result: R = ∞, converges for all x ∈ ℝ
The exponential function has one of the most important power series in mathematics. Since the coefficients are aₙ = 1/n!, the ratio test gives us lim|aₙ₊₁/aₙ| = lim|n!/(n+1)!| = lim|1/(n+1)| = 0. Therefore, the radius of convergence is R = 1/0 = ∞, meaning the series converges for all real numbers. This makes e^x an entire function, and the series provides an exact representation of the exponential function for any value of x. The partial sums S_n = Σ(k=0 to n) x^k/k! approximate e^x with increasing accuracy as n increases.
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