Name: testing series with area under curves
Uploaded: 2026-06-08T09:38:57.969Z
Duration: 1 min 23 s
Channel: JoVE

积分判别法通过将无穷级数的和与由对应于该级数的函数所定义的连续曲线下的面积进行比较，来确定该无穷级数是收敛还是发散。当直接对离散项求和过于困难或不切实际时，此方法至关重要。例如，荧光棒的亮度会随着时间推移逐渐减弱，其光强呈指数级衰减。从第一小时最亮开始，荧光棒的亮度逐渐下降，形成一个每小时能量值的无限序列，难以直接求和。在图表中，每小时释放的能量以垂直条形表示——初始较高，随后迅速变短。一条平滑曲线恰好通过每个条形右上角，将能量释放速率建模为连续函数。此处适用积分判别法，因为该能量辐射函数在时间上为正、连续且单调递减。曲线下面积表示荧光棒释放的总能量。根据数值的变化规律，该面积由一个收敛于有限总值的反常积分定义，因此该级数也收敛。

积分判别法

积分判别法是一种通过将无穷级数与反常积分进行比较，来判断其收敛性或发散性的方法。当级数的项难以直接相加，但符合某个相关连续函数的取值规律时，该方法尤为有用。该判别法无需逐项累加无穷多个离散项，而是通过研究代表相同递减规律的曲线下面积来进行判断。荧光棒为此概念提供了一个有用的示例。激活后，它最初具有很高的亮度，随后随时间推移逐渐减弱。如果记录下每一小时所释放的能量，这些每小时的数值将构成一个无穷数列。将所有这些数值相加，将得到一个表示随时间释放的总能量的无穷级数。在图表中，每个小时的能量值可以用一条垂直的竖条表示。竖条在起始时最高，随着荧光棒亮度减弱而逐渐变短。一条经过每个竖条右上角的平滑曲线，可将能量发射速率建模为连续函数。该连续函数使得可以利用积分对这一序列进行分析。当相关函数在所考虑的区间上为正值、连续且单调递减时，可应用积分判别法。这些条件与荧光棒模型相符，因为荧光棒发射的能量始终保持正值，随时间平滑变化，并随时间推移而逐渐减弱。曲线下方的面积表示荧光棒释放的总连续能量。该面积由一个在无限区间上的反常积分描述。如果该积分收敛于一个有限值，则对应的无穷级数也收敛；如果积分发散，则级数同样发散。在本例中，由于荧光棒的能量输出迅速下降，且曲线下的面积为有限值，因此按小时计算的能量值所构成的无穷级数是收敛的。

https://cloudfront.jove.com/files/ftp_upload/40615/40615_article_thumb_40615.jpeg

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

The Integral Test

积分测试

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, those hourly values make an infinite sequence. Adding them gives an infinite series for the total energy released.

On a graph, each hourly value can be shown as a vertical bar. The bars are tallest at the start and get shorter as the glow stick dims. A smooth curve through the top-right corner of each bar models the energy emission rate as a continuous function. That function lets the series be studied with integration.

The Integral Test works when the related function is positive, continuous, and decreasing on the interval being used. The glow stick model fits these conditions because the energy stays positive, changes smoothly, and decreases over time. The area under the curve represents the total continuous energy released by the glow stick. This area is an improper integral over an infinite interval. If the integral converges to a finite value, the matching infinite series also converges. If the integral diverges, the series diverges too. In this example, the energy output drops quickly enough that the area is finite, so the infinite series of hourly energy values converges.

Testing Series with Area Under Curves

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...

convergence of sequences

Bounded Sequences and the Limit Test

A sequence is an ordered list of numbers created one term at a time. In math analysis, students often ask whether the terms get closer and closer to a single real number as the index grows. When that happens, the sequence converges, and the number it approaches is called the limit. A graph can make this idea easier to see. As the sequence continues, its plotted points move toward a horizontal line. A simple example is the sequence of reciprocals of the natural numbers. Its terms get smaller as...

introduction to infinite series

Convergence and Divergence of Infinite Series

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

Partial sums help show how an infinite series can approach a finite total. An infinite series is made by adding the terms of an infinite sequence. Even though the adding never stops, some series still settle near a fixed value. This idea can model real processes where each step becomes smaller, such as a bouncing ball. A ball dropped from 1 meter rises to half its previous height after each bounce. After the first bounce, it reaches 0.5 meters. After the next bounce, it reaches 0.25 meters.

the integral test

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

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...

bessel function of order zero

Modeling Ripple Decay with Bessel Functions

Bessel functions help model ripple decay in waves with radial symmetry. A simple example is the circular wave that spreads outward after a stone drops into a still pond. The disturbance starts at the center and moves away in all directions. As the wavefront grows, the same energy spreads over a larger circumference. That causes the wave amplitude, or height, to decrease with distance from the center. Simple sine and cosine functions do not describe this change well because their amplitudes...

real-world applications of power series

Pendulum Motion and the Small-Angle Approximation

Pendulum motion can be simplified with the small-angle approximation and a power series. A simple pendulum follows Newton’s Second Law in rotational form, which links the net torque on the bob to its angular acceleration. That relationship leads to a second-order differential equation. The equation is nonlinear because the angular acceleration depends on the sine of the displacement angle. To make the model easier to solve, the sine function is expanded with a Maclaurin series. A Maclaurin...

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 the Integral Test uses a related continuous function and improper integral to decide whether an infinite series converges or diverges.

Testing Series with Area Under Curves | 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 Series with Area Under Curves

Description

Transcript