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

幂级数为数学函数提供了一种替代表示方法，该表示仅在输入值处于特定范围内时收敛。根据级数的不同，存在三种收敛情况：级数可能仅在其中心点收敛，或对所有实数收敛，或在某一有限范围内收敛。该范围称为收敛区间。收敛半径是级数中心到该区间边界的固定距离 R。考虑函数 1 除以 1 减去 x，其幂级数以零为中心。应用比值判别法可得到收敛区间。因此，该级数在负一到一之间的数值范围内收敛。此处，从中心到边界的距离为一，故收敛半径为一。然而，比值检验法在边界处无法得出结论，这些端点必须单独检查。在 x 等于1时，该级数发散。在 x 当其等于负一时，级数振荡且不收敛。因此，收敛区间不包含两个端点，仅为负一到一之间的开区间，在此区间内幂级数表示该函数。

收敛区间与收敛半径

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.

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Sequences and Series

数列与级数

KEY TERMS AND CONCEPTS

关键术语与概念

收敛区间与半径

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

introduction to sequences

Zeno’s Paradox and Geometric Sequences

Zeno’s paradox shows how geometric sequences can describe motion that keeps getting smaller. In the example, a man walks toward a door but only covers half of the remaining distance each time. That creates a sequence of steps: one-half, one-quarter, one-eighth, and so on. This pattern is a geometric sequence because each term is half of the previous term. It can be written as a_n = 1/2^n, where n is the step number starting at 1. The terms shrink in a regular way, even though the sequence...

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introduction to infinite series

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Infinite series are sums that continue without end, and their behavior depends on the pattern of their partial sums. A partial sum is the running total made by adding the first few terms of the series. If these running totals move toward one fixed number, the series converges. If they do not settle near a finite value, the series diverges. A water tank example shows how convergence can happen through repeated halving. Suppose a smart valve releases half of the remaining water every minute. In...

partial sums and series convergence

Bouncing Ball Partial Sums in Series

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the integral test

Testing Series with Area Under Curves

Testing series with area under curves helps determine whether an infinite series converges or diverges. The Integral Test compares a series to an improper integral. It is useful when the terms are hard to add directly but match a related continuous function. The idea is to replace many discrete terms with the area under a curve. A glow stick gives a clear example. After it is activated, it starts bright and then slowly loses intensity over time. If the energy released each hour is recorded,...

comparison tests

Testing Series with Known Behavior

Comparison tests help determine whether an infinite series of positive terms converges or diverges. Instead of finding the sum directly, these methods compare the series with another series whose behavior is already known. That comparison gives a clear answer about whether the total stays finite or grows without bound. The direct comparison test works when each term of an unknown positive series can be matched to a second series term by term. If the unknown terms are less than or equal to the...

alternating series and absolute convergence

alternating-series-and-absolute-convergence

Damped Springs and Alternating Series

Damped spring motion can be modeled with alternating series. A mass on a vertical spring moves above and below a central equilibrium point. In an ideal spring, the motion would keep the same amplitude. In a damped spring, resistive forces such as air resistance or internal friction make each swing smaller over time. This motion is often written as a sinusoidal function with an exponential decay factor. The sine term describes the repeating back-and-forth motion. The decay factor lowers the...

the ratio test

Compound Interest and Series Growth

Compound interest grows an investment by adding interest to both the original amount and the interest already earned. With a 4% annual compound interest rate, each year’s balance becomes 1.04 times the previous year’s balance. That pattern creates a geometric sequence, with 1.04 as the common ratio or repeated multiplier. Mathematicians use the Ratio Test to check whether an infinite series has a finite sum or keeps growing without limit. The test compares consecutive terms in the series and...

power series and their properties

Power Series as Function Approximations

Power series can represent a function as an infinite sum of terms with powers of a variable. The terms are usually written around a central value, called the center. That center is shown as a, so the powers become (x − a)n, where n is a non-negative integer. The coefficients in the series control the behavior of the function. When the center is zero, the series becomes a power expansion in x. It then looks much like an infinite polynomial. This form can model complex functions with high...

interval and radius of convergence

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...

bessel function of order zero

Modeling Ripple Decay with Bessel Functions

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Pendulum Motion and the Small-Angle Approximation

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taylor series

Matching Functions with Taylor Coefficients

Taylor series match a smooth function near a chosen center by using its derivatives. The center is the point where the function is analyzed, and the goal is to build a power series that follows the function’s local behavior. In this expansion, each coefficient is chosen so the series and the function have the same value and the same derivatives at the center. Start with a power series centered at x = a. It has the form Σn=0∞ cn(x − a)n. When x = a, every term with (x − a) becomes zero, so only...

convergence of taylor series

Taylor Series Approximation and Error

Taylor series use polynomials to approximate smooth functions near a chosen point. This makes hard-to-calculate functions easier to work with in science and engineering. A common example is an oscillatory voltage in an alternating current, or AC, circuit. Taylor’s Theorem gives the rule for building these approximations and checking how accurate they are. A Taylor polynomial of order n matches a function’s value and its derivatives up to order n at a central point a. Each derivative adds more...

applications of taylor series

Maclaurin Series for Cosine Motion

Maclaurin series turn a smooth function into an infinite polynomial centered at zero. A Maclaurin series is a Taylor series with the center at zero, so every derivative is taken at that point. That setup often makes the coefficients easier to find. These series are useful in applied mathematics and physics because they replace complicated functions with polynomials that are easier to analyze and compute. One standard example is the cosine function. Its Maclaurin series is cos x = 1 - x^2/2! +...

binomial series

Power Series for Fractional Exponents

The binomial series shows how to expand powers like (1 + x)^m into a power series. It extends the binomial theorem, which works for positive integer exponents. The series also applies to fractional and negative exponents, so it reaches beyond the finite case. It comes from the Maclaurin series for f(x) = (1 + x)^m, where m is any real number. The expansion begins as 1 + mx + m(m - 1)x^2/2! + m(m - 1)(m - 2)x^3/3! + ... . In compact form, it is written as (1 + x)^m = sum from k = 0 to infinity...

key terms and concepts

Learn how radius and interval of convergence show where a power series works, and why the endpoints must be tested separately.

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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