Matpic
mathematics, in pictures
Concept

Conditional convergence

The convergence of a series whose terms add to a finite total only because positive and negative terms cancel, while their absolute values sum to infinity. Riemann showed that such a series can be rearranged to sum to any number at all.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same terms, with the signs alternating. The partial sums of 1 - 1/2 + 1/3 - 1/4 + …, out to 24 terms. They close on 0.69315 from both sides at once, and the gap between consecutive sums is the next term, so the answer is trapped.

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

analysis · Harmonic series Area out to infinity, for three powers. Left: the curves 1/√x, 1/x, 1/x² from x = 0 to 10, with the region beyond x = 1 shaded under the lowest. Right: the area from 1 to T for each, on logarithmic scales, for T up to 10^6. 1/√x keeps growing, 1/x keeps growing, 1/x² levels off at 1.

An endless region with a finite area

A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.

analysis · The integral

Named alongside it

The objects these essays reach for when they reach for this one.

ConvergenceDivergenceHarmonic seriesLimitLogarithmAbsolute convergenceAlternating seriesCommutativityHarmonicsIntegralPiRearrangement

All concepts