Some things blow up. They run away, chain-reaction-fashion, exceeding all limits, like 1 + 2 + 4 + 8 + 16 + .... Or they march steadily out of control like 1 + 1 + 1 + 1 + .... Think about a landslide on an infinite mountain, or compound interest running on forever. In contrast, other things are well-behaved. They stay nice, no matter how much rope one gives them, like 1 + 0.1 + 0.01 + 0.001 + ... which clearly just adds up to 1.111... = 10/9. That's like a decelerating car coming gently to a stop, or any other natural self-limiting process.
Now for the fun part: what's in between, on the bleeding edge of convergence? Take a look at a progression like 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... , the sum of reciprocals. There's growth, but at an ever-decreasing rate. Yes, after ten contributions the pot is up to a bit over 2.9, seemingly moving along at a good pace. But a hundred terms only reaches to just shy of 5.2, and after a thousand the total snails along to not quite 7.5 — and then it really slows down! Throw in millions and billions of further reciprocals, and the sum only crawls up to a grand result of a couple dozen. What? This scarcely looks like runaway growth at all; it's more of a snooze than an explosion.
Yet no matter how big a number we name, add enough terms and the sum will exceed it. It moves slower than molasses in midwinter, slower than any power of the term count — but nevertheless it moves, beyond all bounds. That's an example of logarithmic divergence. Multiplication turns into addition: throw an order of magnitude more donations into the collection plate and the total only grows by a constant (roughly). But grow it does, forever.
Saturday, May 13, 2000 at 09:29:45 (EDT) = 2000-05-13
(correlates: DependentVariables, Forrest Gump --- The Movie, Body Tides, ...)