Harmonic Series is Divergent/Proof 1

Theorem
The harmonic series:


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

diverges.

Proof

 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace{1}_{s_0} + \underbrace{\frac 1 2 + \frac 1 3}_{s_1} + \underbrace{\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$

where $\displaystyle s_k = \sum_{i \mathop = 2^k}^{2^{k+1} \mathop - 1} \frac 1 i$

From Ordering of Reciprocals, $\forall m < n: \dfrac 1 m > \dfrac 1 n$, so each of the summands in a given $s_k$ is greater than $\dfrac 1 {2^{k+1}}$.

The number of summands in a given $s_k$ is $2^{k+1} - 2^k = 2 \times 2^k - 2^k = 2^k$, and so:


 * $s_k > \dfrac{2^k}{2^{k+1}} = \dfrac 1 2$

Hence the harmonic sum satisfies the following inequality:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n = \sum_{k \mathop = 0}^\infty \left({s_k}\right) > \sum_{a \mathop = 1}^\infty \frac 1 2$

The rightmost expression diverges, from the $n$th term test.

The result follows from the the Comparison Test for Divergence.

Historical Note
This proof 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 Johann Bernoulli, and a short time after that by Jakob II Bernoulli, after whom it is usually (erroneously) attributed.