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.