The same series, two universes

What is happening

Fix a prime base p and look at the partial sums of the geometric series:

Sn = 1 + p + p² + ⋯ + pn−1 = (pn − 1) / (p − 1).

Under real analysis, this sequence grows without bound — after all, the terms pk themselves run off to infinity. The real distance from any fixed point can only increase.

In the p-adic view, the story flips. The rule is different: a number is "small" when it is divisible by a high power of p. Since pn is the extreme case of that, the sum Sn sits at a p-adic distance of only p−n from the number

L = 1 / (1 − p) = −1 / (p − 1).

When p = 2, this gives L = −1: the famous 1 + 2 + 4 + 8 + ⋯ = −1. For p = 3, L = −1/2. And so on.

Why the digits agree from the right

In any prime base p, the infinite sum 1 + p + p² + p³ + ⋯ places the digit 1 in every position. That sum equals 1/(1 − p) = L, so the p-adic expansion of the limit is L = …1 1 1 1 — all 1s, no matter which prime. (For p = 2 that coincides with the expansion of −1; for p = 3 with −1/2; and so on.)

Meanwhile Sn = (pn − 1) / (p − 1) has exactly n digits equal to 1 at the right end, followed by zeros: Sn = …000 1 1 … 1. Compared digit by digit, the n rightmost digits of Sn already coincide with L. Each step buys us one more matching digit — and that is exactly what |Sn − L|p = p−n is measuring.

What this little walk shows

Convergence is relative to the metric. The same sequence can diverge in one space and be Cauchy in another. ℚ has two natural completions — one per notion of "size" — and ℝ and ℚp live side by side, with no hierarchy between them. That is the starting point of Ostrowski's theorem, of the p-adic numbers themselves, and of the adelic viewpoint that stitches them all together.

Coming up: post 1 on the main blog (Portuguese) formalises the arithmetic of left-infinite digits and states Ostrowski; playground 02 (coming soon) lets you build −1 digit by digit.