Matpic
mathematics, in pictures
Generator

Partial sums of sin x

A generator in the analysis library, called 60 times across 10 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.

taylor 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

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

ln(1 + x) past its radius, with a denominator allowed

ln(1 + x) past its radius, with a denominator allowed. ln(1 + x), its Taylor sum of degree 8, and its Padé approximants of order 2 and 4. At x = 3 the Taylor sum is out by 5.95e+2 and the highest-order approximant by 2.97e-4.

The poles of the approximants, lined up along the cut

The poles of the approximants, lined up along the cut. The complex plane around ln(1 + x) with the poles of its Padé approximants of order 4 and 8, every one of them on the ray to the left of −1 where the logarithm is cut.

The same coefficients, spent two ways, at x = 3

The same coefficients, spent two ways, at x = 3. Errors at x = 3 for the Padé approximant [n/n] and the Taylor sum of degree 2n of ln(1 + x), for n from 1 to 10, on a logarithmic scale. At n = 10 they are 5.8e-10 and 1.3e+8.

A series that gets better, then worse

A series that gets better, then worse. The error of Euler's series against the number of terms kept, at x = 0.05 and 0.1 and 0.2. At 0.05 the error falls to 1.1e-8 at 20 terms and then climbs without limit. At 0.1 the error falls to 1.8e-4 at 10 terms and then climbs without limit. At 0.2 the error falls to 1.8e-2 at 5 terms and then climbs without limit.

Sums of a series that converges nowhere

Sums of a series that converges nowhere. The function E(x) = ∫₀^∞ e^(−t)/(1 + xt) dt on [0, 0.5] with the partial sums of its Taylor series of 2, 4, 8 terms. Every sum follows the curve near nought and leaves it; the longest is the closest near nought and falls away the hardest.

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.

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 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.

 Analysis

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.

 Analysis

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.

 Analysis

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.

 Analysis

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.

 Analysis

The best nodes have no formula

Interpolating through n points magnifies any error in the data by at most the Lebesgue constant of the points. Chebyshev's points keep it near (2/π)·log n; stretching them to the ends of the interval brings it within two hundredths of the best possible; and the best possible points, characterised in 1978 by having every bump of the error curve the same height, have never been given a formula.

 Analysis

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.

 Analysis

The error that keeps coming back to its worst

Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.

 Analysis

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.

 Analysis

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.

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