Harmonic Series is Divergent/Proof 2

Theorem

The harmonic series:

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

diverges.

Proof

Observe that all the terms of the harmonic series are strictly positive.

From Reciprocal Sequence is Strictly Decreasing, the terms are decreasing.

Hence the Cauchy Condensation Test can be applied, and we examine the convergence of:

\(\ds \sum_{n \mathop = 1}^\infty 2^n \frac 1 {2^n}\)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty 1\)

This diverges, from the $n$th term test.

Hence $\ds \sum \frac 1 n$ also diverges.

$\blacksquare$

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.