< **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). |