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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—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—the sum of all absolute swing lengths—is also finite, the series is absolutely convergent.