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.

`e~~[(1+1/n)^n]_(n=100000)=2.718268237`

Another Expansion

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.

`e~~[(1+n)^(1//n)]_(n=0.000000001)=2.718281827`

The graph of `y=(1+n)^(1//n)` is as follows:
graph of y=(1+n)^(1//n)

(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:

`e=1+1+1/2(1)^2+1/6(1)^3+1/24(1)^4+...`

We can write this as:

`e=sum_(n=0)^oo(1/(n!))`

This series converges to give us the answer correct to 9 decimal places using 12 steps:

`e~~sum_(n=0)^12(1/(n!))=2.718281828`

Brothers' Formulae

Recently, new formulae have been developed by Brothers (2004) which make the calculation of e very efficient.

`e=sum_(n=0)^oo(2n+2)/((2n+1)!`

We only need 6 steps for 9 decimal place accuracy:

`e=sum_(n=0)^6(2n+2)/((2n+1)!)=2.718281828`

Graphical Demonstration of e

The area under the curve `y=1/x` between 1 and e is equal to `1` unit2.
Graph of area under the curve y=1/x between 1 and e

Reference

Brothers, H.J. 2004. Improving the convergence of Newton's series approximation for e. College Mathematics Journal 35(January):34-39..

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

 

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Short URL for this Page

Save typing! You can use this URL to reach this page:

intmath.com/vale

Math Lessons on DVD

 

Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos

Loading...
Loading...