Number · the refusal to resolve

Cross Out Every Nine

The harmonic series adds forever. Delete every fraction whose bottom number contains a chosen digit — and the endless sum quietly stops.

Add up 1 + ½ + ⅓ + ¼ + ⅕ + …, one over every whole number, and the total never settles. It climbs past 10, past 100, past any number you name — slowly, like a coastline going out of sight, but it never stops. Nicole Oresme proved this around 1350, and it is the standard example of a sum whose terms shrink to nothing yet whose total is infinite.

Now cross out some of the terms. Not at random: strike out every fraction whose denominator, written out, contains the digit 9 — so 1/9 goes, and 1/19, 1/90, 1/1900, all of them. Add up only what's left. You would expect a slightly smaller infinity. Instead the sum converges — it lands on a finite number, 22.9206766…, and stays there. This is Kempner's series (1914), and it is not a rounding artifact. Pick a digit and watch it settle.

A minority? No — almost everything.

The first reflex is that you have removed maybe a tenth of the numbers, so the sum should still be infinite, just thinner. But that intuition is calibrated on small numbers. Ask instead: among the numbers with exactly D digits, what fraction contain no 9 at all? Each digit dodges the 9 with probability at best 9/10, and the odds compound. By the time the numbers are long, avoiding the 9 everywhere is a near-miracle.

So for large numbers you keep almost none of them — and the few you keep are enormous, with tiny reciprocals. Kempner's own argument makes the bound concrete: the survivors with D digits number fewer than 8·9^D−1, each smaller than 1/10^D−1, so their reciprocals add to less than 8·(9/10)^D−1. Sum that over all lengths and the whole series is below 8·10 = 80 — finite, in one line, before you know its value. (The true value, 22.92, sits comfortably under the crude 80.)

You cannot get there by adding.

Knowing the sum is finite is not the same as finding it — and finding it by brute addition is hopeless. The survivors thin out so slowly that the running total barely moves. Set it going:

It stalls. Summing the reciprocals of all 43,046,720 no-9 integers below a hundred million — every single one — reaches only 13.2776, still more than nine short of the answer. The finite value instead comes from a different idea: group the survivors by how many digits they have, and carry from each length to the next not just the sum of reciprocals but the sums of reciprocal powers. Appending a digit then turns one length's totals into the next's, exactly. That is Baillie's method (1979) — and it is what the digit readout at the top of this page is running, live, in your browser.

Forbid a rarer pattern, and the sum grows.

Nothing above is special to single digits. Forbid any fixed string of digits — say the numbers must not contain 99 anywhere — and the same thing happens: convergence, to a finite value. But now the survivors are more plentiful (only the rare 99-containing numbers are gone), so the sum is larger. The rarer the forbidden string, the bigger the total — yet always finite, right up until the string is so rare you are keeping essentially everything, and the sum swells back toward the infinite harmonic series it came from.