With s_{n}=1+1/2+1/3+...+1/n, the harmonic series

lim_{n} s_{n}= 
1  + 
1/2  + 
1/3  + 
1/4  + 
1/5  + 
1/6  + 
1/7  + 
1/8  + 
1/9  + 
1/10  + 
1/11  + 
1/12  + 
1/13  + 
1/14  + 
1/15  + 
1/16  + 
... 

> 
1  + 
1/2  + 
1/4  + 
1/4  + 
1/8  + 
1/8  + 
1/8  + 
1/8  + 
1/16  + 
1/16  + 
1/16  + 
1/16  + 
1/16  + 
1/16  + 
1/16  + 
1/16  + 
... 

= 
1  + 
1/2  + 
1/2  + 
1/2  + 
1/2  + 
... 

diverges  but so slowely, that a numerical experiment does not show that.
Even if a machine would have been adding terms at a rate of
10^{9} seconds and would have started 15 billion years ago,
(about 10^{17} seconds), the value of the sum would still be about Log(10^{26})
which is less then 60.
A similar discrepancy between the mathematical fact and experience can
be observed in the
Petersburg paradox or with recurrence theorems in thermodynamics.
