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mathematics, in pictures
Series

Harmonic series — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · analysis
  2. 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.

    part 2 · analysis
  3. The powers of 2 in 1 to 12: one number, 8, stands alone. The whole numbers from 1 to 12, each with a bar whose height is the power of 2 dividing it. A single number has the tallest bar, which is why the harmonic number H(12) has an even denominator and an odd numerator.

    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.

    part 3 · analysis
  4. 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.

    part 4 · analysis

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