A 14th century proof of the divergence of the harmonic series

Nicole d’Oresme was a philosopher from 14th century France. He’s credited for finding the first proof of the divergence of the harmonic series. In other words, he authored the first proof we know of for the fact that $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$ is infinite.

His proof is very simple, so much so that I think probably someone with more talent for educating than me could probably teach it to middle school students. I think that’s really cool.

Formally, the sum we’re talking about is the sum of $1/k$ for all positive integers $k$ . What d’Oresme did was proving that for any number $C$ , you can find a finite $k$ so that the sum of the first $k$ terms (i.e. $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{k-1} + \frac{1}{k}$ ) in the original sum is larger than $C$ . If the first $k$ terms sum to more than $C$ , then the entire sum must be larger than $C$ as well (because there are no negative numbers in the sum). That means the sum is larger than any finite number and, therefore, it has to be infinite.

But how do you find the right $k$ that will make your sum large enough? Well, d’Oresme used the following fact: if you take all the $1/k$ numbers with $k>=n$ and $k<2n$ , then you have $n$ numbers that are all larger than $\frac{1}{2n}$ . The sum of all $n$ of them must be larger than $\frac{n}{2n} = \frac{1}{2}$ .

The above proves that the first number (i.e. with $k>=1$ and $k<2*1$ ) has to be larger than $\frac{1}{2}$ . From $2$ to $3$ (i.e. $k>=2,k<4$ ) the sum should also be larger than $\frac{1}{2}$ . The numbers from $4$ to $7$ also sum more than $\frac{1}{2}$ . In other words, the sum of numbers from any power of $2$ to the next power of $2$ should sum over $\frac{1}{2}$ . It’s easy to see from this that the first $2^{2C}$ numbers in the series should sum something that’s larger than $C$ .

In other words, the $k$ such that the sum of the first $k$ terms is higher than $C$ is $k=2^{2C}$ . And that, in a nutshell, is Nicole d’Oresme’s proof of the divergence of the harmonic series.

Nicole d’Oresme studied a bunch of cool topics. I think he, as pretty much all other medieval scientists, is an under-appreciated historical figure. I like reading about medieval scientists and their works, so I might post other interesting things from them in the future.

Do bear in mind that I’m not a historian, so any historical commentary I include in my posts about medieval scientists should be taken with a grain of salt.

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