Harmonic Series is Divergent/Proof 3

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Theorem

The harmonic series:

$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n$

diverges.


Proof

We have that the Integral of Reciprocal is Divergent.

Hence from the Integral Test, the harmonic series also diverges.

$\blacksquare$



Note

The integral test works in both directions.

That is, it can also be used to show that Integral of Reciprocal is Divergent based on Harmonic Series is Divergent.

This could lead to a circular proof.

Hence, if the integral test is used here, it should not be used to prove Integral of Reciprocal is Divergent.

Instead, use for instance the definition of the natural logarithm as integral of reciprocal.


Historical Note

The proof that the Harmonic Series is Divergent was discovered by Nicole Oresme.

However, it was lost for centuries, before being rediscovered by Pietro Mengoli in $1647$.

It was discovered yet again in $1687$ by Johann Bernoulli, and a short time after that by Jakob II Bernoulli, after whom it is usually (erroneously) attributed.

Some sources attribute its rediscovery to Jacob Bernoulli.


Sources