Text Box: Management
Text Box: Travels and Geopolitics
Text Box: Theories and facts that nobody can afford not knowing (or at least to have a vague idea)
Text Box: Harmonic Series 

It is amazing how slowly the harmonic series increases to reach the infinite. 
The harmonic series diverges: 
 
A very elegant and easy demonstration:
After the two first terms, 1 and ½, we divide the series in segments of length 2n.  Under this partition we can easily see that the sum of each segment is greater than   
  For the first two terms obviously 1+1/2>1/2. 
Next 21 terms:  
                                                          
next 22 terms:
                                                                  
next 23 terms:
                                                                                        
                                                                                                   8   times
next 24 terms:
                                                      …                     …..
										 16  times

This partition can go up to infinite and each segment will be greater than. 
So   
 Let’s see how slowly the harmonic series goes to infinite:
Let S(n) the area of the rectangle (αβγδ) and P(n) the area of (βγζε).
We can easily see that:



Adding up from n=1 to n we get:
  or 
1+1/2+1/3+........1/n<1+ln(n)
From the latest inequality we have that the first sum of the first million terms of the harmonic is less than 15. The first billion terms add up to less than 22!!.