Name: testing power series convergence bounds
Uploaded: 2026-06-08T09:38:24.166Z
Duration: 1 min 29 s
Channel: JoVE

A power series provides an alternative representation of mathematical functions that converge only when the inputs remain within a specific range.Depending on the series, there are three cases of convergence. The series might converge only at its center, or for every real number, or for a finite range.This range is called the interval of convergence. The radius of convergence is the fixed distance R from the center of the series to the boundary of that interval.Consider the function one over one minus x with its power series centered at zero.Applying the Ratio Test gives the interval of convergence. So the series converges for the values between negative one and one. Here, the distance from the center to the boundary is one, giving a radius of convergence of one.However, the Ratio Test is inconclusive at the boundaries. These endpoints must be checked separately.At x equals one, the series diverges. At x equals negative one, the series oscillates and does not converge. Therefore, the interval of convergence excludes both endpoints and is between negative one and one, where the power series represents the function.

interval and radius of convergence

A power series is a mathematical representation of a function as an infinite sum of terms involving powers of a variable. Such series converge only for specific input values, making it essential to determine the range over which the series produces valid results. This leads to the concepts of radius and interval of convergence, which define where the series behaves meaningfully.The radius of convergence describes the distance from the center within which the power series converges. For a general power series centered at a, this can be written as:\begin{equation*}\sum\limits_{n=0}^{\infty} c_n (x - a)^n\end{equation*}A standard method to determine this radius is the ratio test. The limit of the ratio of consecutive terms is:\begin{equation*}L = \lim_{n \to \infty} \left| \jfrac{c_{n+1}(x - a)^{n+1}}{c_n (x - a)^n} \right|\end{equation*}This simplifies to:\begin{equation*}L = |x - a| \lim_{n \to \infty} \left| \jfrac{c_{n+1}}{c_n} \right|\end{equation*}For convergence, the condition L &lt; 1 must hold, which gives:\begin{equation*}|x - a| \lim_{n \to \infty} \left| \jfrac{c_{n+1}}{c_n} \right|&lt;1\end{equation*}Rearranging this inequality yields the radius of convergence:\begin{equation*}R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|\end{equation*}The interval of convergence is the set of all real values of x that satisfy:\begin{equation*}|x-a|&lt;R\end{equation*}This defines the open interval (a - R, a + R). However, the endpoints x = a &#xB1; R must be tested separately, since the ratio test is inconclusive at these boundary points.

Interval and Radius of Convergence

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testing power series convergence bounds

Power series converge only for certain input values, so it is important to find the range where they work. The radius of convergence gives the distance from the center where the series still converges. The interval of convergence shows the real x-values that fit inside that range.

A general power series can be written as a sum of terms of the form c_n(x - a)^n, where a is the center. To find the radius of convergence, the ratio test is often used. It compares consecutive terms by taking the limit of their ratio.

For the series, that limit becomes |x - a| times the limit of |c_{n+1}/c_n|. The series converges when this value is less than 1. Rearranging the inequality gives the radius of convergence as R = lim |c_n/c_{n+1}|.

The interval of convergence is then the open interval (a - R, a + R), which comes from the condition |x - a| &lt; R. The endpoints x = a ± R must be checked separately. The ratio test does not decide what happens at those boundary points.

Testing Power Series Convergence Bounds

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Testing Power Series Convergence Bounds | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Testing Power Series Convergence Bounds

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