Convergence
Named by 55 essays across 6 fields — each of them below, with the objects they name alongside it.
Adding up rectangles until they stop being rectangles
The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.
A square wave built entirely out of round ones
Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.
A bell curve assembled out of coin flips
Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.
Getting pi by dropping needles on the floor
Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.
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.
One point's worth of information
A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.
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.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
A point that pulls, and a point that pushes
Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.
A constant that does not care which map
The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.
Where Newton's method goes instead
An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.
The rule that forgets where it came from
A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.
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.
The staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
The average settles and the wobble does not
Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.
A curve with a corner at every point
Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.
Where the coefficients come from
The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.
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 shape that averaging leaves alone
Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.
An average that never settles
The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.
The size of a number with no formula
There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.
How long until it forgets
The essays before this one settle where a chain ends up and how much time it spends there, and none of them asks how long the settling takes. That question has an exact answer, it is a single number, and it is the only thing any practical use of a chain depends on.
The sieve that cannot finish
Sifting out the composites is the oldest method in the subject and it has a ceiling nobody has raised. The reciprocals of the twin primes add to a finite number, so no argument that measures thickness can reach them — and the inclusion–exclusion every sieve truncates goes wildly wrong before it goes right.
An error with an unknown in it
Taylor's theorem does not say a partial sum is close to anything. It says the error is one more derivative evaluated somewhere nobody can name, and everything the theorem is worth comes from what happens when that somewhere is replaced by the worst case.
The centre is a choice
A Taylor series is nearly always written about zero, and nothing about the construction prefers zero. Moving the centre moves the interval the series works on, and moving it repeatedly walks the function into places its first series could never reach.
The points that ruin the fit
A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.
A denominator that reaches past the radius
The Taylor series of ln(1 + x) is useless beyond x = 1 however many terms it is given. The same coefficients spent on a numerator and a denominator converge at x = 3, and at x = 100, because a polynomial cannot imitate a singularity and a quotient of two polynomials can.
A series that converges nowhere
Expand Euler's integral in powers of x and the coefficients are the factorials, so the series converges at no x but nought. Stopped at its smallest term it still computes the integral to within about e^(−1/x) — and every term added after that makes the answer worse.
![Partial sums of x − x²/2 + x³/3 − … on [0, 1]. Partial sums of the power series x − x²/2 + x³/3 − … drawn on the interval from 0 to 1 with the function the series sums to inside the interval, and the values at x = 1 marked.](/f/partial-sums-of-x-x2-2-x3-3-on-0-1-e402dfbd.svg)
A sum read from inside
The series 1 − 1/2 + 1/3 − 1/4 + … adds to log 2, and the reason is not in the series. Its power series equals log(1 + x) inside the interval, and the value at the edge is read off by continuity. Abel's theorem says when that reading is honest — and the series 1 − 1 + 1 − …, which has no sum, is read the same way as a half.
Which curves have a length at all
A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.
The reading that is almost right
Every account of simultaneous choice so far has assumed the payoffs are known to both choosers and known to be known. Replace that with each chooser seeing a private reading off by a little, and a band of equilibria closes to a single point — so the assumption nobody states decides the answer.
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.
An area that never finishes
Newton's method's famous failure is a boundary, and a boundary has no area — a random start misses it with probability one. The real failure is different in kind: a polynomial with small whole-number coefficients whose method has a region of starting points, with area, from which it provably never terminates.
Covering rather than avoiding
Two arguments say no starting guess is safe: the boundary is fractal and some regions are permanently trapped. The repair is not a better guess. It is a fixed list of starting points, computed from the degree alone, from which every root of every polynomial of that degree is found.
Why a quarter of numbers overshoot
About one number in four has proper divisors adding to more than itself. That a proportion exists at all is not automatic — there are sets defined just as simply that have no proportion — and the reason this one does is that abundance is inherited by multiples, and the numbers it is first inherited from are sparse enough to add up.
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 fraction Lambert built for the tangent
The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.
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.
Uniform, except on a small set
The functions xⁿ settle on their limit at every point and never uniformly — the trouble is all in a strip next to 1. Throw the strip away and the convergence is uniform on what is left, however thin the strip. Egorov proved that this always happens on an interval, Lusin proved the matching fact about a single function, and a bump sliding off along the whole line shows why both need a set of finite length to start from.
The ripples that make a series run away
Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.
Every chord slid to the middle
Take a shape, pick a line, and slide every chord that crosses the line at right angles until the line cuts it in half. The area cannot change, the boundary can only get shorter, and the result is symmetric. Do it again about another line, and another, and the shape is squeezed towards a disc — unless the lines are badly chosen, in which case it stops short.
A geometric series whose ratio is a matrix
1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.
A series that converges to minus one
1 + 2 + 4 + 8 + … runs off to infinity, and yet the formula for a geometric series says it should equal 1/(1 − 2) = −1. Measure size by how many factors of 2 a number has, instead of how large it is, and powers of 2 become small: the series converges, and to exactly −1. The same change of ruler explains why every repeating decimal is a fraction, and why repeating binary digits running off to the left are fractions too.
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
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 series that waits on π
Add 1/(n³ sin² n) for n = 1, 2, 3, … and the terms are mostly tiny, except where n is almost a multiple of π and sin n is almost nought. Ten million terms add to 30.3145, four-fifths of it from the single term at n = 355. Whether the sum is finite depends on how closely fractions can approach π — on a number called its irrationality measure — and the best proof available says only that the measure is below 7.1, where the series needs it below 2.5.
The sine, rebuilt from its zeros
A polynomial is a product of factors, one for each root. Euler treated the sine the same way — one factor for each place the wave crosses the axis — and the product he wrote down is correct, though the zeros alone do not justify it. Multiplied out, it hands over the sum of the reciprocal squares, π²/6, and every even power after it.
A sum of factorials that converges at every prime
1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.
The length round an ellipse
An ellipse has an area anyone can write down, πab, and a perimeter nobody can: it is an integral with no expression in the functions of school mathematics. Kepler guessed it, Euler guessed better, and in 1914 Ramanujan wrote down a formula whose error falls like the fifth power of how far the ellipse is from a circle — off by four hundredths of a per cent even for a segment. Gauss had already found the way to compute it exactly, doubling the correct digits at every step.
Turning a slow series geometric
Leibniz's series 1 − 1/3 + 1/5 − 1/7 + … adds up to π/4, and it takes five billion terms to give ten decimal places. Euler found a rule that rewrites it, using exactly the same terms, as a series whose terms halve at every step — and forty of those give thirteen places. Aitken's and Shanks's later rules do better still. None of them works on every slow series, and the reason is that each one is a guess about the shape of the error.
The zeros a partial sum cannot avoid
The exponential is never zero, and every partial sum 1 + z + z²/2! + … + zⁿ/n! of its series is zero n times. Divide those zeros by n and they line up on a single egg-shaped curve, |z e^(1−z)| = 1, found by Szegő in 1924. The curve is where two sizes tie — the exponential the series is reaching for, and the terms it has not yet reached — and the same tie, carried one step further, places every zero to within a few hundredths.
The sum after Leibniz's, made geometric
Leibniz's series adds the reciprocals of the odd numbers with alternating signs and gives π/4. Square the odd numbers and the same pattern gives Catalan's constant, 0.915965…, about which almost nothing is known — not even whether it is a fraction. A formula of Ramanujan's turns its slow series into one whose terms shrink by a factor of four, which makes a thousand decimals cheap, and the decimals and continued fraction then say exactly how much digits can settle and where they stop.
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.
Root two, raised to itself for ever
Stack √2 to the power √2 to the power √2 without end and the tower settles at exactly 2. Solve the equation for a tower equal to 4 and √2 appears again, and that answer is false. The bases that give a limit run from e^(−e) to e^(1/e), the slope at the crossing decides everything, and a sum over labelled trees turns out to be the same number.
Named alongside it
The objects these essays reach for when they reach for this one.
LimitApproximationPiGeometric seriesHarmonic seriesRadius of convergenceTaylor seriesContinuityCounterexampleConvergence rateIndependenceIteration