Fastest converging method for calculating e

Question

Does anyone know what the fastest converging equation is for calculating e? I have found many different equations when I have been searching google, but I was wondering which ones can yield the most digits of e in the fewest terms.

Thank you.

Answer 1

I don't really know how fast this approach is, but as a programmer I find it easy to code/debug and also very aesthetic as a mathematician. If you are familiar with Continued Fractions, you will be very pleased to know that the notation for the important constant $e$ is: $$ e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...]$$

Alternatively, I know that the Taylor series expansion of $e$ as

$$ e^x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots $$ This means that $$e = e^1 = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots $$ You've probably encountered at least the Taylor series expansion already in your research but I know that the 10th convergent of $e$ is $\dfrac{1457}{536} = 2.718283582 \approx e = 2.718281828...$ so the continued fraction method gives 5 decimal places correctly in 10 terms (hopefully this is a useful benchmark for you).

Answer 2

In the same spirit as Xoque55, there is another possible approach based on Pade approximants; if you are not familiar, I suggest you have a look at
http://mathworld.wolfram.com/PadeApproximant.html

For simplicity, I shall use here expansions of the same degree on both numerator and denominator and, for each approximant, "x" will be replaced by the value x=1, just as in Xoque55 answer.

I give you below, the approximate value of "e" as a function of the selected degree

degree 1 gives 3.000000000
degree 2 gives 2.714285714 (19 / 7)
degree 3 gives 2.718309859 (193 / 71)
degree 4 gives 2.718281718 (2721 / 1001)
degree 5 gives 2.718281829 (49171 / 18089)
degree 6 gives 2.718281828 (1084483 / 398959)