Harmonic Series is Divergent/Proof 4

Theorem

The harmonic series:

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

diverges.

Proof

For all $N \in \N$:

$\dfrac 1 N + \dfrac 1 {N + 1} + \cdots + \dfrac 1 {2 N} > N \cdot \dfrac 1 {2 N} = \dfrac 1 2$

Hence, by Cauchy's Convergence Criterion for Series, the Harmonic series is divergent.

$\blacksquare$

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.

Sources

• 1970: A.J.B. Ward: 216. Divergence of the harmonic series (The Mathematical Gazette Vol. 54, no. 389: p. 277)