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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.



The theorem was elaborated in detail by the German mathematician Bernhard Riemann (1826 - 1866).

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