But the 5 has to be a large number to get very close to e, this next one is faster:
But I found this one that is even faster for just a little more arithmetic:
This one is 3 more digits correct for n=5.
I looked online for a list of many different possible power series for e but the first two I listed here are the only ones I found so maybe this one is new. I thought of it as kind of a combination of the first two and it seems to work.
A proof of why it converges faster is that the terms inside the parenthesis are actually the Taylor series for the 5th root of e^x centered around x = 0 evaluated at x=1. And since the terms of that series are getting smaller so much faster than the Taylor series for e^x centered around 0 (because of the 5^i terms in the denominators , less terms must converge to it's limit faster. Then you raise it to the 5th power to get e.
The fives in the denominator of the terms in the parenthesis can be increased to any number as long as the exponent on the parenthesis matches, or the series in the parenthesis can be lengthened and still raised to the same power on the parenthesized expression. So since there are two ways to go I'm not sure which is easier to calculate or whether there is a difference in the rate of convergence but maybe I'll look more into it at some point.The two classical approaches to calculating e that I first listed can thus both be thought of as special cases of the formula I've found. The first uses the first two terms of the nth root series raised to the nth power, the second uses the first root equation to the power of 1.
Actually my professor Dr. Rose recalled a paper he had read by Harlan Brothers and John Knox that discussed different faster approximation's for e and it turns out an improved version of this one I found was found by Harlan Brothers here: http://www.brotherstechnology.com/math/e-formulas.html Formula 25 results from what he calls compressing the series I have here.