Power Series Convergence: How to Find the Radius (R) and Test Endpoints
To find the interval and radius of convergence for an infinite power series $\sum a_n (x - c)^n$, first apply the Ratio Test: evaluate $L = \lim_{n \to \infty} |u_{n+1} / u_n| = |x - c| \cdot \lim_{n \to \infty} |a_{n+1} / a_n|$. Set $L < 1$ to solve for $|x - c| < R$, where $R$ is the Radius of Convergence. Finally, substitute the endpoints $x = c - R$ and $x = c + R$ individually into the series and test convergence using p-series or alternating series theorems.
Key Takeaway Facts
- Every power series centered at c converges in one of three ways: solely at x = c (R = 0), on all real numbers (-∞, ∞) (R = ∞), or on a bounded interval of radius R > 0.
- The Ratio Test is strictly inconclusive when L = 1; it cannot determine convergence at boundary endpoints.
- If a series includes factorials (e.g., n!), the radius of convergence is typically infinite (R = ∞).
- Parentheses ( ) indicate open non-inclusive endpoints, while brackets [ ] indicate closed convergent endpoints.
- Geometric series Σ xⁿ always diverge at endpoints x = ±1 by the nth term divergence test, giving (-1, 1).
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1. The Step-by-Step Algorithm for Power Series Convergence
Frequently Asked Questions
Can a power series have a radius of convergence of zero?
Yes. If the ratio test limit is infinite for all x ≠ c (such as in Σ n! xⁿ), the radius of convergence is R = 0, meaning the series converges only at the center x = 0.
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This guide was researched and drafted by the ToolQix Mathematical Analysis Group and technically verified by Professor of Pure & Applied Mathematics under ToolQix's strict accuracy protocols. Formulas, calculations, and instructions were independently tested against current industry specifications.