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This is a very poor description of this formula for e^x - it isn’t a miracle at all that it converges! It’s pretty simple. The numerators are x^n. The denomina
by lacker 7y ago
This is a very poor description of this formula for e^x - it isn’t a miracle at all that it converges!
It’s pretty simple. The numerators are x^n. The denominators are n!. For any x, the denominators grow faster than the numerators, so it’s no surprise that it converges. The ratio only starts shrinking when n > x but you can see that with high school math.
Somehow all these largish random numbers manage to cancel out almost completely.
No, they aren’t “largish random numbers”, it’s just the ratio between two series, one of which grows asymptotically faster than the other.
- Dylan16807 7y agoConverging isn't a surprise. This giant mishmash somehow converging near zero is a surprise, and getting something with e is even more of a surprise. Nowhere in the post does it express surprise about the fact that each series converges on something. Separately, "largish random numbers" is a perfectly fine way to describe the start of the series, which is the most important part for influencing what it eventually converges on. Somehow despite each series getting increasingly enormous before dropping back toward zero, the convergence point ends up less and less perturbed by those enormous terms.