Harmonic Series is Divergent/Proof 1

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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:

$\forall m, n \in \N_{>0}: m < n: \dfrac 1 m > \dfrac 1 n$

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 $H_{2^m}$ satisfies the following inequality:

\(\displaystyle H_{2^m}\) \(=\) \(\displaystyle \sum_{n \mathop = 1}^{2^m} \frac 1 n\) $\quad$ $\quad$
\(\displaystyle \) \(>\) \(\displaystyle \sum_{n \mathop = 1}^{2^m - 1} \frac 1 n\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^m \left({s_k}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(>\) \(\displaystyle 1 + \sum_{a \mathop = 0}^m \frac 1 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac m 2\) $\quad$ $\quad$


The right hand side diverges, from the $n$th term test.

The result follows from the the Comparison Test for Divergence.

$\blacksquare$


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