Generator

converge

A generator in the analysis library, called 7 times across 1 essays. Below: what it draws at its defaults 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.

At its defaults

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.

`show: "sup"`

The largest gap to the limit, against n. The supremum distance between each member of a sequence of functions and its pointwise limit, plotted against n for two sequences — one whose distance falls to zero and one whose does not.

`show: "tube"`

A tube of width 0.4 about the limit of x / n. A band drawn about the limit function with several members of the sequence over it, showing whether any member lies entirely inside the band and which is the first that does.

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.

a member fits inside the tube exactly when the convergence is uniform ×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
no point of the interval beats the stated largest gap ×1
one to three sequences this family draws ×1
refining the grid finds a larger gap, because the largest is reached nowhere ×1
the grid finds the largest gap, because it is reached at a point ×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 shrinks exactly when the convergence is uniform ×1
the members are given in increasing order ×1
the search for a member inside the tube runs to between 20 and 2000 ×1
the sequence is one this family draws ×1
the sup distance is plotted for between 6 and 80 values of n ×1
the tube has a half-width between 0.02 and 0.6 ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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.

The whole library · What the figures prove