Partial sum

A partial sum is the sum of the first finitely many terms of a sequence, especially a series. The sequence of partial sums tracks how the total changes as more terms are included.

Sequence: An ordered list of mathematical objects, often numbers, indexed by positive integers. A partial sum takes an initial segment from a sequence of terms.
Running total: A cumulative sum updated by adding each new value to the previous total. The partial sums of a sequence form its running total.
Numerical series: A sum whose terms are real or complex numbers, potentially with infinitely many terms. Its finite approximations are computed as partial sums.
Convergent series: A series whose sequence of partial sums approaches a finite limit. Convergence is defined by the limiting behavior of these finite totals.
Series: An expression formed by adding the terms of a sequence, either finitely or infinitely. Partial sums are the finite sums used to define an infinite series.
Recurrence relation: An equation that defines terms using one or more earlier terms. Each partial sum equals the preceding sum plus the next sequence term.
Riemann sum: A finite weighted sum used to approximate the area under a function. Refining partitions produces successive finite sums that approach an integral.
Divergent series: A series whose sequence of partial sums does not approach a finite limit. Its partial sums fail to settle at a finite value.
Summation notation: A compact notation using the Greek letter sigma to represent sums over indexed terms. Sigma notation specifies the terms and bounds in a partial sum.
Telescoping series: A series whose terms cancel in succession when its finite partial sums are expanded. Its partial sums simplify to a small number of uncancelled terms.

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