Zero: The Biography of a Dangerous Idea

by Charles Seife

Like the Zeno sequence and Suiseth’s sequence, all the terms get closer and closer to zero. However, when Oresme tried to sum the terms in the sequence, he realized that the sums got larger and larger and larger. Even though the individual terms go to zero, the sum goes off to infinity. Oresme showed this by clumping the terms together: 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + . . . . The first group clearly equals 1/2; the second group is greater than (1/4 + 1/4), or 1/2. The third group is greater than (1/8 + 1/8 + 1/8 + 1/8), or 1/2. And so forth. You keep adding 1/2 after 1/2 after 1/2, and the sum gets bigger and bigger, and off to infinity. Even though the terms themselves go to zero, they don’t approach zero fast enough. An infinite sum of numbers can be infinite, even if the numbers themselves approach zero. Yet this isn’t the strangest aspect of infinite sums. Zero itself is not immune to the bizarre nature of infinity.