< 1 >
Riemann series theorem
The sum of the terms in a conditionally convergent alternating infinit harmonic series depends on the order in which the terms are added.
Example 1
We start with the alternating harmonic series
Now we divide by 2 and insert a zero in front of each term
By adding the two series term-by-term, we get
Removing the zeroes shows clearly that each of the terms occurs only once, albeith now in a different order
So we obtain exactly the same harmonic series, but it is 
of the original value.
HistoryThe theorem was elaborated in detail by the German mathematician Bernhard Riemann (1826 - 1866). |