Name: damped springs and alternating series
Uploaded: 2026-06-08T09:38:20.592Z
Duration: 1 min 28 s
Channel: JoVE

A mass on a vertical spring oscillates around a central equilibrium point, with its oscillations gradually decreasing over time.This motion is modeled by a damped spring equation, where an exponential function reduces the amplitude of the swings.To analyze this motion, the sine component is replaced by its Taylor series&#x2014;an infinite sum that approximates a function using its derivatives. This substitution transforms the model into an alternating series, where terms switch between positive and negative values.Expanding the summation reveals the individual terms of the motion one by one. The Alternating Series Test then checks for convergence using two conditions.First, the magnitudes of the terms must decrease steadily, similar to how each oscillation of the spring is smaller than the one before.Second, the magnitudes must approach zero, showing the point at which motion stops. When both conditions are met, the alternating series converges to a finite sum, just as the damped spring eventually settles at rest.If the total distance traveled&#x2014;the sum of all absolute swing lengths&#x2014;is also finite, the series is absolutely convergent.

alternating series and absolute convergence

Alternating Series and Absolute Convergence

A mass attached to a vertical spring can exhibit oscillatory motion as it moves above and below a central equilibrium point. In an ideal spring, the oscillations would continue indefinitely with constant amplitude. In a damped spring, however, resistive forces such as air resistance or internal friction gradually reduce the size of each swing. This behavior is often modeled by combining a sinusoidal function, which represents the repeated motion, with an exponential decay factor, which reduces the amplitude over time.To study the mathematical structure of the damped motion, the sine component can be replaced by its Taylor series. A Taylor series expresses a function as an infinite sum of terms based on its derivatives. For the sine function, the resulting expansion contains powers of the variable whose signs alternate between positive and negative.When this expansion is substituted into the damped spring model, the motion can be represented as an alternating series. Each term contributes part of the overall displacement, and the alternating signs reflect the back-and-forth nature of the spring&#x2019;s motion around equilibrium.The Alternating Series Test determines whether such a series converges to a finite value. It requires two conditions. First, the magnitudes of the terms must decrease steadily. This corresponds to the physical behavior of damping, where each oscillation of the spring is smaller than the previous one. Second, the magnitudes of the terms must approach zero. This represents the long-term behavior of the system, in which the mass gradually comes to rest at the equilibrium position.If both conditions are satisfied, the alternating series converges. In the damped spring model, this means the mathematical description approaches a finite value, just as the physical motion settles over time. If the sum of the absolute values of all terms is also finite, then the series is absolutely convergent, indicating that the total accumulated motion remains bounded.

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JoVE Core

Calculus

Unit

Sequences and Series

KEY TERMS AND CONCEPTS

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 amplitude as time passes. To study the math behind this model, the sine function can be replaced by its Taylor series.

A Taylor series is an infinite sum built from a function and its derivatives. For sine, the expansion includes powers of the variable with alternating positive and negative signs. When this expansion is used in the damped spring model, the result is an alternating series. Each term adds part of the total displacement, and the sign changes match the spring’s motion around equilibrium.

The Alternating Series Test checks whether such a series converges to a finite value. It requires two conditions. The term sizes must decrease steadily, which matches the way damping makes each oscillation smaller. The term sizes must also approach zero, which matches the long-term behavior of the mass as it comes to rest at equilibrium.

If both conditions are met, the alternating series converges. For the damped spring model, that means the mathematical description approaches a finite value as the motion settles. If the sum of the absolute values of the terms is also finite, the series is absolutely convergent. That means the total accumulated motion stays bounded.

Damped Springs and Alternating Series

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key terms and concepts

Explore how damped spring motion leads to an alternating series, then use the Alternating Series Test to check convergence and absolute convergence.

Damped Springs and Alternating Series | 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

Damped Springs and Alternating Series

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