partial sum

A partial sum is the running total after you have added only the first few terms of a series and stopped. It is the number on the cash register after the first n items, before the rest are rung up.

The n-th partial sum, written S_n, equals a_1 + a_2 + ... + a_n. These partial sums themselves form a new sequence: S_1, S_2, S_3, ..., where each is the previous one plus the next term. Watching this sequence is how we make sense of an infinite series.

If the partial sums settle down toward a single number as n grows, that limiting value is the sum of the infinite series, and we say the series converges. If they grow without bound or never settle, the series diverges and has no finite sum.