Matpic
mathematics, in pictures
Series

Taylor series — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · analysis
  2. 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.

    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.

    part 2 · analysis
  3. 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.

    part 3 · analysis
  4. 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.

    part 4 · analysis
  5. 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.

    part 5 · analysis
  6. 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.

    part 6 · analysis

All series