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.
+495 I just played with my calculator and I see exactly what you're talking about.
Basically what you're saying is: instead of (1+1/n)^n, doing (1+1/n)^(n+0.5) gives you more accurate digits.
(1+1/100)^100 = 2.7048...
(1+1/100)^100.5 = 2.7183 ... (exactly 2x more accurate digits)
(1+1/1000)^1000 = 2.7169239 ...
(1+1/1000)^1000.5 = 2.71828205... (again 2x more accurate digits)
It's pretty interesting, but I don't know why it would give you more accurate digits. Maybe it is a coincidence?
