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Concept

Radius of convergence

The distance from a power series' centre out to where it stops converging. It equals the distance to the nearest point where the function is not analytic, measured in the complex plane whether or not the function was ever meant to leave the real line, and it can also be read off the growth of the coefficients alone.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

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.

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 · Taylor series 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.

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 · Taylor series 11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350.

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 · Taylor series 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.

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 · Taylor series 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.

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 · Taylor series 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.

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 · Uniform convergence A disc of radius 0.2 and an ellipse of size 1.22 around the same interval. For 1/(1 + 25x²): the interval from −1 to 1 on the real axis, poles at 0 + 0.2i and 0 − 0.2i, the Taylor disc at 0 of radius 0.20, and the Bernstein ellipse with foci ±1 through the poles, with ρ = 1.2198.

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 · Taylor series The 16 zigzag arrangements of 5 numbers. All alternating permutations of 1..5: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.

The tangent counts zigzags

Arrange the numbers 1 to n so that they go up, down, up, down. There are 1, 1, 2, 5, 16, 61, 272 such arrangements, and those are, multiplied by factorials, the Taylor coefficients of tan x and sec x — the odd lengths from the tangent, the even from the secant. Désiré André proved it in 1879 by splitting a zigzag at its largest number, and the split turns out to be the derivative of the tangent written out term by term.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

ConvergenceTaylor seriesApproximationPolynomial approximationSingularityAnalytic functionAnalytic continuationComplex numbersError analysisPade approximantRemainderAbel summation

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