partial sum

Since you cannot add infinitely many numbers all at once, you stop at some stage and add only what you have so far. The result of stopping after the first N terms is the N-th partial sum. As N grows, these snapshots trace out the progress of the addition, and the fate of the whole series is decided by where these snapshots go.

Precisely, for a series sum a_n the N-th partial sum is s_N = a_1 + a_2 + ... + a_N, a perfectly ordinary finite sum and hence a genuine number for each N. The partial sums form a sequence (s_N), and the series converges exactly when this sequence has a finite limit. Thus the study of series is, technically, the study of one particular sequence built from the terms.

A useful identity links the two viewpoints: a_N = s_N - s_{N-1} recovers each term from consecutive partial sums. This is why a series and its sequence of partial sums encode exactly the same information, just packaged differently — one as increments, the other as accumulated totals.