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Doing infinitely many things — page 4

Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.

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.

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