10.4
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.
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.
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.
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