Harmonic series — the series
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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.
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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.
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The sum that steps over every whole number
The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.
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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.
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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.
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Two sign patterns that land together
At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.
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What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
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A rule that counts lets the sum escape
Strike out every number containing a 9 and the harmonic series converges, to 22.92. Keep only the numbers with as many 7s as 3s and it diverges — slowly, like the square root of the number of digits. Ask for two such balances and it still diverges; ask for three and it converges. The dividing line is Pólya's theorem that a random walk comes home in one and two dimensions and not in three.
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Which powers of x are enough
Every continuous function on an interval is nearly a polynomial, and it seems that all the powers of x should be needed for that. They are not. The even powers will do, and so will the primes, though the primes run out at a rate of one number in ln n. The squares will not: however many are combined, they stay 0.0905 away from x itself. The dividing line, found by Müntz in 1914, is whether the reciprocals of the powers add up to infinity.
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A table that never reaches π
Fill a square table with 1/1, 1/2, 1/3 along the rows, each row starting one place further on. Every row is a piece of the harmonic series, and yet the table stretches no vector by as much as π. At a thousand rows the stretch is 2.42, at a trillion it is near 2.94, and one integral explains both the number and the failure to arrive.