Calculating the Value of e
There are several ways to calculate the value of e. Let's look at the historical development.
Using a Binomial Expansion
If
n is very large (approaches infinity) the value of
approaches e.
The largest that Scientific Notebook can handle is about n = 100,000 and this is only correct to the 4th decimal place.
Another Expansion
As n becomes very small,
approaches the value of e.
We can obtain reasonable accuracy with a very small value of n.
The graph of y = (1 + x)1/x
is as follows:
(There is actually a "hole" at x = 0. Can you understand why?)
Newton's Series Expansion for e
The
series expansion for e
is
Replacing x with 1, we have:
We can write this as:
This series converges to give us the answer correct to 9 decimal places using 12 steps:
Brother's Formulae
Recently, new formulae have been developed by Brothers (2004) which make the calculation of e very efficient.
We only need 6 steps for 9 decimal place accuracy:
Graphical Demonstration of e
The area under the curve
between 1 and e
is equal to 1
unit2.
Reference:
Brothers, H.J. 2004. Improving the convergence of Newton's series approximation for e. College Mathematics Journal 35(January):34-39. Available at http://www.brotherstechnology.com/docs/icnsae_(cmj0104-300dpi).pdf.
Back to 
Logarithms with Base e.
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