Re: Average life of distribution equipment
It was that great mathematician Leonhard Euler who discovered the number e and calculated its value to 23 decimal places. It is often called Euler's number and, like pi, is a transcendental number (this means it is not the root of any algebraic equation with integer coefficients). Its properties have led to it as a "natural" choice as a logarithmic base, and indeed e is also known as the natural base or Naperian base (after John Napier).
There is the remarkable property that if the function ex (known as the exponential function and also denoted as "exp(x) ") is differentiated with respect to x, then the result is the same function ex. The proof of this can be seen in many textbooks on elementary calculus.
To answer this question we need to evaluate
lim
n ? ?
(1 + 1/n)n.
This quantity turns out again to be e - the same base value with the property that the gradient of the graph is unity at x = 0.
Now limn ? ? (1 + 1/n)n can be expanded very nicely using the trusty old Binomial Theorem. We find that
e = 1+ 1
--------------------------------------------------------------------------------
1!
+ 1
--------------------------------------------------------------------------------
2!
+ 1
--------------------------------------------------------------------------------
3!
+ ... + 1
--------------------------------------------------------------------------------
r!
+ ....
This series is convergent, and evaluating the sum far enough to give no change in the fourth decimal place (this occurs after the seventh term is added) gives an approximation for e of 2.718.
