Generator

xⁿ at 5 values of n, and the limit

A generator in the analysis library, called 46 times across 8 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

converge is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

Thomae's function as a limit of tents, at n = 3 and 8

Where each function jumps by at least 1/5

A set with no interval in it, half the line long, and continuous functions falling to it

A limit of limits: cos(m!πx) to the power 2n, for m = 2 and 3

Upper sums of the spike functions, and of their limit

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

so sin 2π1x and sin 2π2x differ by at least 1 somewhere ×90
the mean square of sin 2π1x − sin 2π2x over [0, 1] is 1 ×90
the grid never beats the stated modulus of sin(2πkx) at k = 1 ×40
the grid never beats the stated modulus of sin(2πx + k) at k = 1 ×40
the grid never beats the stated modulus of xᵏ at k = 1 ×40
x − x²/2 + x³/3 − …: 1 terms stay within the bound ×30
x − x³/3 + x⁵/5 − …: 1 terms stay within the bound ×30
members 7 and 13 share an arc, so they are within 0.765 everywhere ×28
the n = 3 approximant is 1/1 at 0/1 ×28
at 0/2 every member is 1 ×10
the sum of 1 terms is within 0.5000 of the function everywhere on [0, 1], x = 1 included ×10
a member later than the last one picked lies in the arc at stage 0 ×9
the two members agree at node 0/8 ×9
1 − 2x + 3x² − 4x³ + … at x = 1/2 is the function it is said to sum to ×1
1 − x + x² − x³ + … at x = 1/2 is the function it is said to sum to ×1
1 − x + x² − x³ + …: the largest gap on [0, 1] never drops below 1/2 ×1
2 arctan t − t solves y′ = cos(t + y) ×1
4 to 12 nodes ×1
a member fits inside the tube exactly when the convergence is uniform ×1
a well-chosen polynomial of the same degree does better than the iteration ×1
all four kinds of fraction occur ×1
and between the nodes they are nearly 2 apart ×1
and comes within two hundredths of it ×1
and N is the first such member ×1
at 2π/3 the increment shrinks like a square root ×1
at an irrational point the approximants settle towards 0 ×1
at the golden point the averaged exponent lies near three-quarters, well clear of one-half and one ×1
at x = 1 the partial sums stay away from the function's value there ×1
at π the remainder after the slope shrinks like h to the three-halves ×1
between 16 and 400 members ×1
between 4 and 10 halvings ×1
between the multiples of 1/m! the members fall to 0 ×1
between two and seven steps, each from 1 to 11 ×1
between two and six levels of cutting ×1
between two and six member numbers, each a whole number up to 400 ×1
each column of the picture settles on the limit ×1
each member is within 1/√n of |x|, worst at 0 ×1
each point is passed by the bump and left behind ×1
eps is read only by the covering view among these ×1
every level covers each point exactly once ×1
every point settles on the limit ×1
every shifted wave has the same modulus at every k ×1
every step has slope between −1 and 1 ×1
finer polygons sit closer to the solution ×1
from N on, the members are inside the band off the strip ×1
near the ends the differentiated series runs off to minus infinity ×1
no point of the interval beats the stated largest gap ×1
once a point is within the tolerance it stays within it ×1
one half holds at least half the members ×1
one member is within the tolerance on the whole interval ×1
one to four members are drawn, from inside the table ×1
one to three approximants, each with n from 2 to 40 ×1
one to three sequences this family draws ×1
one to three values of m, each from 1 to 4 ×1
R vanishes at π, where every term sin(n²π) is zero ×1
refining the grid finds a larger gap, because the largest is reached nowhere ×1
some point is still far from the limit ×1
steps is read only by the climbing view ×1
term counts up to between 8 and 60 ×1
the antiderivative is at most 2/n³ and reaches it ×1
the bars run to between 20 and 80 ×1
the circle of phases is cut into 4 to 16 arcs ×1
the differentiated series at 0 alternates between 2 and 0 ×1
the fullest arc holds at least its share ×1
the gap to the previous pick is within the previous arc's bound ×1
the Gauss sum vanishes exactly when p and q are both odd ×1
the grid finds the largest gap, because it is reached at a point ×1
the height of each member is 1/n ×1
the intervals have total length below ε ×1
the largest gap for x / n falls with n ×1
the largest gap for xⁿ does not fall with n ×1
the largest gap for xⁿ never drops below where it started ×1
the largest gap is below a constant over √n ×1
the largest gap shrinks as n grows ×1
the largest gap shrinks exactly when the convergence is uniform ×1
the measured distance between partial sums respects the tail bound ×1
the measured square-root coefficient matches the Gauss sum on both sides ×1
the members are given in increasing order ×1
the members considered number 500 to 20000 ×1
the members fall at every point ×1
the members run to between 10 and 60 ×1
the partial sums at x = 1 do not settle ×1
the points jumping by at least 1/5 are the fractions with q ≤ 5 ×1
the power family's modulus rises with k ×1
the removed length is the geometric sum ×1
the search for a member inside the tube runs to between 20 and 2000 ×1
the second frequency is one more than a multiple of the node count, so the two agree at every node ×1
the sequence is one this family draws ×1
the series is one this family draws ×1
the series sums to x inside the interval ×1
the sine family's modulus reaches 2 ×1
the stated gap between two members is the one on the grid ×1
the steepest slope grows with every term ×1
the steepest slope is at most π times the sum of (ab)ᵏ ×1
the steepest slope of each member is n ×1
the stretch δ is between 0.02 and 0.2 ×1
the sup distance is plotted for between 6 and 80 values of n ×1
the symmetric difference quotient settles on −1/2 ×1
the table of gaps runs to between 4 and 12 ×1
the threshold is 1/k for k from 2 to 12 ×1
the time runs to between 1 and 6 ×1
the tolerance is between 0 and a half ×1
the tolerance is between 0.02 and 0.3 ×1
the tube has a half-width between 0.02 and 0.6 ×1
the upper sums of a function with finitely many spikes fall like 1/N ×1
the view is one the family draws ×1
the weight on partial sums equal to 1 is 1/(1 + x) ×1
the weights add to one, the unshown tail included ×1
the worst point moves in towards 0 as the steps go on ×1
two or three series ×1
two to five levels ×1
two to five step counts, increasing, up to 256 ×1
two to six term counts, increasing, up to 60 ×1
two values of x between 0.3 and 0.995, increasing ×1
which is the function the series sums to ×1
x − x²/2 + x³/3 − … at x = 1/2 is the function it is said to sum to ×1
x − x²/2 + x³/3 − …: the largest gap on [0, 1] falls away ×1
x − x³/3 + x⁵/5 − … at x = 1/2 is the function it is said to sum to ×1
x − x³/3 + x⁵/5 − …: the largest gap on [0, 1] falls away ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

A limit can jump at every fraction

A sequence of continuous functions can settle, point by point, on a function that is discontinuous at every rational number. It cannot settle on one that is discontinuous everywhere — the indicator of the rationals needs two limits in a row, and Riemann's integral cannot follow the second. The line between the two is Baire's theorem, and it measures smallness by gaps rather than by length.

Analysis

A limit that forgets to be continuous

Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.

Analysis

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.

Analysis

Close in height and nowhere close in slope

Uniform convergence carries continuity to the limit and carries integrals to the limit. It carries slopes nowhere. The functions sin(n²x)/n flatten to nothing while their slopes grow without bound; smooth curves converge uniformly to a corner; a Fourier series converges and its differentiated series diverges. What does carry a slope is uniform convergence of the slopes themselves — and the reason is that integration averages wiggles away while differentiation multiplies them.

Analysis

Settling in order settles everywhere at once

A sequence of functions can settle at every point and never settle uniformly — the largest gap stays put while each point escapes it. Dini found the circumstances in which that cannot happen: continuous functions, a continuous limit, every member above the next, and a closed interval. Under those four conditions convergence at each point is convergence everywhere at once, and Weierstrass's iteration for the square root becomes a sequence of polynomials converging uniformly to |x|.

Analysis

The slope a Gauss sum leaves behind

Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.

Analysis

The subsequence that has to exist

Every bounded list of numbers has a part that settles down. A bounded list of functions need not: the waves sin 2πkx never come within 1.76 of one another. One extra condition — that no member may change faster than a bound they all share — restores the guarantee, and it is the reason a differential equation with a continuous rule has a solution at all.

Analysis

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 whole library · What the figures prove