Name: bouncing ball partial sums in series
Uploaded: 2026-06-08T09:38:48.438Z
Duration: 1 min 23 s
Channel: JoVE

An infinite series can demonstrate how adding infinitely many terms leads to a definite sum.It helps explain physical processes that involve smaller and smaller actions, such as the motion of a bouncing ball.Consider a ball dropped from a height of one meter. After each bounce, the ball rises to exactly half the height of the previous drop.These heights form a specific type of infinite series.To calculate the sum of the maximum heights, mathematicians use partial sums, or running totals. The first partial sum includes only the initial drop.The second adds the height of the first bounce, giving a total of one point five meters. Each subsequent partial sum adds the next, smaller distance.As more terms are added, the total sum is represented by an infinite series. Because each term shrinks rapidly, the sum approaches but never exceeds a finite value of two meters.Because the partial sums reach a finite value, the series is convergent. However, in other cases, like the infinite series of natural numbers, the total grows without a bound, making it a divergent series.

partial sums and series convergence

An infinite series is formed by adding the terms of an infinite sequence. Although the addition continues without end, some infinite series approach a definite finite value. This idea is useful for modeling physical processes in which each successive action becomes smaller, such as the motion of a bouncing ball that rises to a fraction of its previous height after each bounce.Consider a ball dropped from a height of one meter. After the first drop, it rises to half of that height, or 0.5 meters. After the next bounce, it rises to half of that height, or 0.25 meters. This pattern continues, with each bounce reaching half the height of the previous one.To analyze the total of these maximum heights, mathematicians use partial sums. A partial sum is a running total of the first several terms of a series. The first partial sum includes only the initial height of 1 meter. The second partial sum adds the first bounce height, giving 1.5 meters. The next partial sums add 0.25, then 0.125, and so on.As more bounce heights are added, the running totals increase but get closer and closer to a fixed value. In this case, the sum approaches 2 meters and never exceeds it. Because the partial sums approach a finite limit, the infinite series is convergent.This example shows that infinitely many terms do not always produce an infinite total. When the terms decrease rapidly enough, their sum can remain finite. In contrast, some series do not converge. For example, the infinite series formed by adding the natural numbers grows without bound. Since its partial sums do not approach a finite value, it is a divergent series.

Partial Sums and Series Convergence

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Education

JoVE Core

Calculus

Математический анализ

Unit

Единица

Sequences and Series

Последовательности и ряды

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Частичные суммы и сходимость рядов

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 pattern keeps going, and each bounce is half the height of the one before it.

Mathematicians use partial sums to track the total of these maximum heights. A partial sum is the running total of the first terms in a series. The first partial sum is 1 meter. The second adds the first bounce height and becomes 1.5 meters. Later partial sums add 0.25, then 0.125, and continue in the same way.

As more bounce heights are included, the running totals keep rising but get closer to a fixed limit. In this example, the sum approaches 2 meters and does not go past it. Because the partial sums approach a finite value, the infinite series is convergent. This shows that infinitely many terms do not always make an infinite total.

Some series do not converge. For example, the infinite series made by adding the natural numbers grows without bound. Its partial sums do not approach a finite value. That makes it a divergent series.

Bouncing Ball Partial Sums in Series

introduction to sequences

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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 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

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Testing Series with Known Behavior

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alternating-series-and-absolute-convergence

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the ratio test

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taylor series

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Taylor Series Approximation and Error

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key terms and concepts

See how partial sums track a bouncing ball’s heights and show why some infinite series converge to a finite limit while others diverge.

Bouncing Ball Partial Sums in 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

Bouncing Ball Partial Sums in Series

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