Infinite Series and Geometric Series

An infinite series adds the terms of a sequence $a_1, a_2, a_3, \dots$ without stopping. Nobody can do infinitely many additions, so the value of the series is defined through the finite sums that lead up to it.

The terms and the partial sums are two different sequences. A series converges when its partial sums settle on a number, and the terms only matter through the totals they produce. In Example 1 below, the terms go to 0 while the partial sums go to $\frac{8}{3}$.

All three series below have terms that approach 0, like the series in Example 1. Their partial sums still behave differently, and only two of them settle on a number.

A geometric series has a constant ratio $r$ between successive terms, so each term is $r$ times the one before. With first term $a$, its partial sums have a closed form. Subtract $rS_n$ from $S_n$, and every term but two cancels.

When $|r| < 1$, $r^n \to 0$ and the partial sums approach $\frac{a}{1 - r}$. Now suppose $a \ne 0$. When $|r| > 1$, $|r^n|$ grows without bound, so the partial sums don't settle. With $r = 1$ the partial sums are $na$, and with $r = -1$ they alternate between $a$ and 0.

The formula needs the first term of the series, whatever its index. A series that starts at $n = 2$ still sums to the first term over $1 - r$.