Matpic
mathematics, in pictures
Concept

Divergence

The failure of a sequence or a sum to settle on a value, whether by growing without bound or by never coming to rest. It comes in two kinds worth separating: growth without bound, and oscillation that never settles on anything.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

A sum whose terms vanish and whose total does not

Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.

analysis · Harmonic series 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 The sieve as a product, and the sum over the primes. The whole numbers up to 60, with those built only from 2, 3, 5 marked — the numbers the product of three geometric series multiplies out to. Beside them, the sum of the reciprocals of the primes, which grows without bound.

The sieve written as a product

Multiply out one geometric series for each prime and every whole number appears exactly once, as a single term. That identity turns a statement about factorisation into a statement about convergence, and it is where the analytic study of the primes begins.

number · Prime distribution Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes 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.

ConvergenceHarmonic seriesLimitLogarithmConditional convergenceHarmonicsPiAbsolute convergenceAlternating seriesAsymptoticBlock stackingCommutativity

All concepts