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Geometric series — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

    The sum that fits in one square

    Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

    part 1 · analysis
  2. n / 2ⁿ held under a geometric series. A bar for each term of the series n / 2ⁿ with a decaying geometric curve above them, the curve lying above every bar from term 2 onwards.

    The series everything else is measured against

    A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.

    part 2 · analysis
  3. Runs of doubling length in 1/n^2, and the geometric series that bounds them. A bar for the total of each run of terms of 1/n to the 2, with an outlined bar above it for the bound obtained by replacing every term in the run with its largest, the bounds forming a geometric series.

    The repair at the boundary

    Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.

    part 3 · analysis
  4. Powers of a matrix that shrink in the end. Three curves of the norm of the n-th power of a two-by-two matrix against n on a logarithmic axis: one decays steadily, two rise to peaks of about 18 and 7 before decaying.

    A geometric series whose ratio is a matrix

    1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.

    part 4 · analysis
  5. 1 + 2 + 4 + … in two notions of size. Two sets of points against the number of terms up to 20 on an axis whose gridlines are powers of 2: the partial sums' ordinary size rising, their 2-adic distance from −1 falling.

    A series that converges to minus one

    1 + 2 + 4 + 8 + … runs off to infinity, and yet the formula for a geometric series says it should equal 1/(1 − 2) = −1. Measure size by how many factors of 2 a number has, instead of how large it is, and powers of 2 become small: the series converges, and to exactly −1. The same change of ruler explains why every repeating decimal is a fraction, and why repeating binary digits running off to the left are fractions too.

    part 5 · analysis

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