Convergence and Divergence of Infinite Series

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 s...

An infinite series is the sum of an infinite sequence of terms, where the addition continues forever. It is defined by the limit of its partial sums.

Imagine a water tank with a smart valve programmed to release exactly half of the remaining water every minute.

In the next minute, half of what’s left drains out—that’s one-quarter. Th...