# Harmonic Series is Divergent/Proof 3

## Contents

## Theorem

The harmonic series:

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

## 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

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 9.3$