Partial sum
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
A series that waits on π
Add 1/(n³ sin² n) for n = 1, 2, 3, … and the terms are mostly tiny, except where n is almost a multiple of π and sin n is almost nought. Ten million terms add to 30.3145, four-fifths of it from the single term at n = 355. Whether the sum is finite depends on how closely fractions can approach π — on a number called its irrationality measure — and the best proof available says only that the measure is below 7.1, where the series needs it below 2.5.
Turning a slow series geometric
Leibniz's series 1 − 1/3 + 1/5 − 1/7 + … adds up to π/4, and it takes five billion terms to give ten decimal places. Euler found a rule that rewrites it, using exactly the same terms, as a series whose terms halve at every step — and forty of those give thirteen places. Aitken's and Shanks's later rules do better still. None of them works on every slow series, and the reason is that each one is a guess about the shape of the error.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvergencePiAlternating seriesApproximationContinued fractionsDiophantine approximationGeometric seriesIrrationality measureSeriesSmall divisors