The harmonic series
satisfies the necessary condition of convergence
for the series of real or complex terms:
Nevertheless, the harmonic series diverges. It is seen if we first group the terms with parentheses:
Here, each parenthetic sum contains a number of terms twice as many as the preceding one. The sum in the first parentheses is greater than , the sum in the second parentheses is greater than ; thus one sees that the sum in all parentheses is greater than . Consequently, the partial sum of first terms exceeds any given real number, when is sufficiently big.
The divergence of the harmonic series is very slow, though. Its speed may be illustrated by considering the difference
(see the diagram). We know that increases very slowly as (e.g. ). The increasing of the partial sum is about the same, since the limit
is a little positive number
which is called the Euler constant or Euler-Mascheroni constant.
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