Harmonic series
Named by 25 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
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.
There is no last prime
Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.
How long until every one turns up
Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.
The sum that fits in one square
Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.
When to stop looking
Candidates arrive one at a time in a random order. Each must be accepted or rejected on the spot, with no going back and no way to know what is still to come. The best possible rule is to look at about a third of them and then take the first one that beats everything seen — and it works about a third of the time, however many there are.
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.
The area that names the number
The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.
Add the odds from the end
Watch a sequence of independent events and try to stop exactly on the last one that happens. Add up the odds of the events from the end backwards until the total reaches one, and stop at the first success from there. That rule is the best possible for any probabilities whatever, and the secretary problem is the special case in which the chances are one over the position.
The series everything else is measured against
A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.
The repair at the boundary
Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.
The one that hardly ever comes up
Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.
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.
The surface a random gluing makes
Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.
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.
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.
One extra person on one side
In a random market of a thousand a side, whoever proposes gets about their seventh choice and whoever receives gets about their hundred-and-fortieth. Add one person to one side and the advantage of proposing all but disappears: the shorter side does well and the longer side badly, whichever side proposes, and most people are left with exactly one stable partner.
No integrand sits on the border
The area under 1/x out to infinity is infinite and the area under 1/x² is finite, so it is natural to look for the dividing line. There is none. Divide any divergent integrand by its own running integral and it still diverges, more slowly; divide any convergent one by the square root of its tail and it still converges, more slowly. Paul du Bois-Reymond proved in 1873 that no single integrand can separate the two families — and the slow ones are slow beyond imagining.
A wait that ends and has no average
Draw a number, then keep drawing until one beats it. Half the time the very next draw does. The wait is certain to end, it is usually short, and its average is infinite — and the chance of still waiting after n draws is exactly 1/(n + 1), whatever the numbers are drawn from, because symmetry is the only thing the calculation uses.
Seven times half of π, and then not
The area under sin x / x, out to infinity, is π/2. Multiply in sin(x/3)/(x/3) and the area is still π/2; multiply in the waves for 5, 7, 9, 11 and 13 and it is still π/2, exactly. Add the wave for 15 and the area falls short — by about 2.3 × 10⁻¹¹. Nothing about the curves changes at the eighth factor. What changes is that 1/3 + 1/5 + … + 1/15 has just passed 1.
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.
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.
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.
A hundred prisoners and one shuffle
A hundred prisoners each search fifty of a hundred boxes for their own number, one at a time, with no communication once it starts. If any fails, all are executed. Choosing at random gives them a chance of 8 × 10⁻³¹. There is a rule that gives them 31%, and no rule can do better. Every prisoner's individual chance is still one half: what the rule changes is not how often each succeeds but how those successes line up.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvergenceConvergence rateExpectationLimitLogarithmDivergenceGeometric seriesPrimesIntegralPiCounting argumente, the number