# 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:

(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` unit^{2}.

### 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.)

Go to: Online algebra solver

### 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!

### 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