e = 1 + 1/1! + 1/2! + 1/3! +............ The sum of this series can provide the value of e. Using Chebychev economization of the series a fast and accurate solution can be obtained. A simple algo will be as follows

(i) sum = 1.;

(ii) n=1;

(iii) fact = n; (stop when fact is less than accuracy)

(iv) term = 1./fact ;

(v) sum = sum + term;

(vi) n= n+1;

(vii) go to (iii)

Euler numbers $E_n$ are integers that arise in the expansion of the hyperbolic secant function around the origin, where

\[

sech(x) = \mathop{sum}\limits_{n = 0}^{\infty} \frac{x^nE_n}{n!}.

\]

Using the CDF from Mathematica, the attached plot was obtained.

http://demonstrations.wolfram.com/PeriodicityOfEulerNumbersInModularArithmetic/

To find n digits in e (base for natural logs) using Mathematica 9, try

N[E,n]

For example,

N[E,50] produces

2.7182818284590452353602874713526624977572470937000

Indeed James, using Mathematica we may reach excellent value for e. However, in terms of analytic formula, which one is the easiest?

e = 1 + 1/1! + 1/2! + 1/3! +............ The sum of this series can provide the value of e. Using Chebychev economization of the series a fast and accurate solution can be obtained. A simple algo will be as follows

(i) sum = 1.;

(ii) n=1;

(iii) fact = n; (stop when fact is less than accuracy)

(iv) term = 1./fact ;

(v) sum = sum + term;

(vi) n= n+1;

(vii) go to (iii)

Stop when term is less than accuracy. Make this correction.

The series mentioned in the earlier answer is called 'e'. The definition of natural logarithm is based on this irrational number, and the logarithmic series came to be defined very naturally.

I have not understood the question. Why should we try to define 'e' differently?

There is the well known exercise lim (1+1/n)^n for large n. Probably very slow!

Well, in literature we have different ways to compute for the value of e, the same for the factorial of a number. In this sense, it would be good to know which one is more accurate and can be obtained more easily and quickly..