Concept

Almost surely

With probability one: true of every outcome except a set of outcomes whose total chance is nought. A random series converging almost surely may still diverge for some sign patterns, and those exceptions can be uncountably many while carrying no probability at all.

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

12 harmonic series with random signs, each settling on its own sum. Partial sums of the harmonic series with each sign chosen by a fair coin, for several independent runs, plotted against the number of terms on a logarithmic scale, beside the all-plus and alternating sign patterns.

A coin in front of every term

Put all plus signs in front of 1, 1/2, 1/3, … and the sum runs off to infinity; alternate them and it settles on log 2. Toss a fair coin for each sign instead, and the sum settles — every time, on a different number. Where it tends to settle has a smooth, flat-topped shape, and at the value 2 that shape takes a height that agrees with one eighth to forty-two decimal places and is not one eighth.

analysis · Harmonic series

Where a coin-signed geometric series lands, for λ = 1/3, 1/2, 1/√2, 1/φ. Histograms of the exact distribution of the sum of plus or minus λ to the k, one panel per value of λ: a dust of separated pieces below one half, a flat block at one half, and smooth-looking overlapping shapes above.

A coin in front of every power

Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.

analysis · Harmonic series

Named alongside it

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

Binary expansion Probability density Bernoulli convolution Cantor set Characteristic function Convergence Fractal dimension Geometric series Harmonic series Random walk Self-similarity Variance

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