Hello Alan: I wonder if you saw this observation that I noticed about the digits of e. Any insights? Thanks:
As you know, one way to calculate "e" is this very simple exponential formula:
(1 + 1/n)^n. For large n, such as 10^6, the number of accurate digits of e is proportional to the exponent, or 6 accurate digits in this case. But, when you use n + 1/2 as an exponent (or 1,000,000.5 in this case), the number of accurate digits of e goes up as 2n, or twice as many accurate digits of e!!. The question is: why? Appreciate any insights. Many thanks.
+33665 I did see this. I don't know that I have an explanation as such though! It's just another series that converges to e (in the limit as n tends to infinity the (1 + 1/n)^(1/2) term tends to 1). The expression (1 + 1/n)^n starts off as a poor approximation to e for small values of n, and the (1 + 1/n)^(1/2) term moves it in the right direction, while gradually reducing towards 1 itself as n gets bigger.
