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A proof of the divergence of the harmonic series
Alexander Frolkin
8 August 2001
We start off with the following definition of the Riemann Zeta
function:
We turn our attention to the integrand:
Some manipulation of the series yields
Plugging back into the Zeta function,
Substitute
in the integral1, so that
Plugging back into the definition,
This leads to the more common definition of the Riemann Zeta function,
If we put
, the function reduces to the harmonic series --
We can find the value of
using the integral definition. So,
we have
To evaluate the integral
substitute
Hence,
Since these are improper integrals, in the Riemann sense, we need to
take limits. Now,
We have
Since
We have
Hence,
completing the proof that the harmonic series diverges.
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Alexander Frolkin
2001-08-16