Euler's number e - and a bit of pi

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A few pages about Euler’s number e … and then a bit aboutπ

The limit of (1+1/n)ⁿ as n->infinity

$\displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,{{\left( {1+\frac{1}{n}} \right)}^{n}}$

The function we get from the limit of (1+x/n)ⁿ as n->infinity

$\displaystyle f(x)=\underset{{n\to \infty }}{\mathop{{\lim }}}\,{{\left( {1+\frac{x}{n}} \right)}^{n}}$

The function e^x

We may use the compound formula, (1+1/n)ⁿ as n->infinity as a rather inefficient way to calculate e. That is explored in the following page.

Calculating e using the compound formula

We found one way to calculate e here above. As mentioned, it is rather inefficient though. You can find better methods here:

Finding 10000 decimals of e (This will lead you away from this part of this site.)

In the following pages we will explore what happens if the input to the compound formula is an imaginary value, and if we can make the output become –1. In the process we will discover that we can find the value of pi using the same formula that we just used to find the value of e, showing the extreme close relation between those two numbers.

Up a level : Calculus and Analysis
Previous page : Differentiation, derivatives
Next page : Power Series

Last modified: May 9, 2026 @ 13:12