Generator

Partial sums of sin x

A generator in the analysis library, called 26 times across 4 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.

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.

At its defaults

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.

`show: "remainder"`

What the degree-5 sum costs, and what the bound claims. The error of the degree-5 Taylor polynomial of sin x against x, on a logarithmic scale, with Lagrange's bound drawn above it. The bound exceeds the error by a factor of 1.8 at the right-hand end.

`show: "xi"`

Where the point Taylor's theorem cannot name actually is. The position of the mean-value point ξ, as a fraction of the way from 0 to x, for the degree-1, 2, 3, 4 remainders of eˣ. Each settles on 1/(n+2) as x shrinks.

`show: "radius"`

A smooth function whose series stops working. 1/(1 + x²) with partial sums of degree 4, 12, 28. Inside |x| < 1 more terms improve the fit; outside it they make it worse, and the function is smooth throughout.

`show: "centre"`

1/(1 + x²), expanded about 1.2. 1/(1 + x²) with Taylor sums of degree 2, 6, 14 about x = 1.2 rather than about zero. The interval they converge on reaches 1.562 either side of the centre.

`show: "discs"`

The disc a Taylor series converges on. The complex plane with 2 singularities of 1/(1 + z²) and the disc of convergence about 1 centre, each disc reaching exactly to the nearest of them.

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.

the polynomial passes through node 1 ×21
the degree-1 sum matches to order 1 at the expansion point ×7
the degree-2 sum about 1.2 matches the function to order 2 there ×6
as x shrinks, the point settles at 1/(n+2) for degree 1 ×4
the degree-2 sum about -1.4 matches the function to order 2 there ×3
between 3 and 41 nodes ×2
a degree between 1 and 20 ×1
an amplification factor is never below one, since the nodes reproduce themselves ×1
and by the largest count drawn it is orders of magnitude worse ×1
and it dwarfs anything in the middle of the interval ×1
between one and five degrees, each between 1 and 8 ×1
between one and four degrees ×1
between one and four singularities, each a point of the plane ×1
between one and six centres ×1
between three and fourteen node counts, each between 3 and 25 ×1
Chebyshev nodes keep the error small across the whole interval ×1
each new centre lies inside the previous disc, so its coefficients are known there ×1
each node is the projection of its angle ×1
equally spaced nodes never amplify less than Chebyshev ones ×1
every term count is a whole number between 1 and 40 ×1
inside the new interval more terms help ×1
inside the radius, more terms help ×1
ln(1+x) is centred where it is defined ×1
ln(1+x) is expanded inside its own radius ×1
no centre sits on a singularity ×1
no singularity lies strictly inside a disc ×1
outside it they hurt ×1
outside it, more terms hurt ×1
the bound is a bound ×1
the centre is inside the drawn domain ×1
the Chebyshev end gaps are several times smaller than the middle one ×1
the equally spaced constant grows with every step ×1
the error never exceeds the bound Lagrange's form puts on it ×1
the evenly spaced nodes have one gap throughout ×1
the function is one the figure knows ×1
the function is one whose derivatives the figure knows ×1
the interval drawn is inside the function's radius ×1
the interval drawn is not empty ×1
the interval is between 0.4 and 8 wide ×1
the mode draws the functions whose derivative is monotone ×1
the mode of the taylor family is one of sums, radius, remainder, xi, centre, discs, interp, lebesgue, nodes ×1
the nodes are equally spaced or Chebyshev ×1
the point moves steadily in one direction as x grows ×1
the radius is the distance to the nearest singularity ×1
the smallest gap is at one of the ends ×1
the tail of the series is the difference between the function and its partial sum ×1
the unknown point lies strictly between nought and x ×1
with equally spaced nodes the worst error is out near the ends ×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

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 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 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.

The whole library · What the figures prove