Definition
Approximation as infinite series
Approximation as infinite continued fraction
Or, writing it in one line become
Very condense form of continued fraction is show that the continued fraction of has regular pattern of where is increasing even numbers.
MS excel function of Euler number is =EXP(1)
Now you can
download the Spreadsheet companion of this tutorial
and try PI() for different accuracy
For example:
Accuracy 
Exp as ratio 
Error (computed from PI of MS Excel) 
0.1 


0.001 


0.00001 


0.0000001 


This tutorial is copyrighted .
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html