Introduzione alle serie infinite

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. Then one-eighth, then one-sixteenth. This pattern continues forever, with each amount being half of the one before.

At first, it feels like a paradox. If you are always removing something, it seems the tank should never truly be empty.

There’s always a tiny, microscopic drop remaining. But as time goes on, that remaining amount becomes so small it effectively disappears.

To track this, mathematicians use partial sums, a running total of all the water that has left the tank.

If these running totals approach a fixed number, the series converges. In this case, the sum approaches the total volume of the tank.

However, some series don't settle on a finite value. Compounded wealth, for example, multiplies itself by a fixed interest rate every interval, causing the total to grow endlessly in a divergent series.