Harmonic Series is Divergent/Proof 1

Theorem
The harmonic series:


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

diverges.

Proof

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

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

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

Hence the harmonic sum:
 * $\displaystyle \sum_{n=1}^\infty \frac 1 n = \sum_{k=0}^\infty \left({s_k}\right) > \sum_{a=1}^\infty \frac 1 2$

the last of which diverges, from the Nth Term Test.

The result follows from the the Comparison Test for Divergence.

Historical Note
This proof is generally attributed to Jakob II Bernoulli.

However, some sources suggest that it may have been discovered some 400 years earlier by Nicole Oresme.