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 `(1+1/n)^n`approaches e.
The largest that Scientific Notebook can handle is about n = 100,000 and this is only correct to the 4th decimal place.
As n becomes very small, `(1+n)^(1//n)` approaches the value of e.
We can obtain reasonable accuracy with a very small value of n.
The graph of `y=(1+n)^(1//n)` is as follows:
(There is actually a "hole" at n = 0. Can you understand why?)
Newton's Series Expansion for e
The series expansion for e is `e^x=1+x+1/2x^2+1/6x^3+1/24x^4+...`
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:
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 `y=1/x` between 1 and e is equal to `1` unit2.
Brothers, H.J. 2004. Improving the convergence of Newton's series approximation for e. College Mathematics Journal 35(January):34-39..
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