The Harmonic Series

The Harmonic series is the infinite series whose sum is given by

The harmonic series diverges. This is quite easy to prove.

Thusis unbounded sois divergent.

In factis in many cases divergent for manyfor example if

S is the set of prime numbers

S is the set of multiples of any number.

The series diverges very slowly. For example, the sum of the firstterms is less than 100. This is because the partial sums of the series have logarithmic growth:

as can easy be proved using the integral test.

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