“So the soul of immensity dwells in minutia. And in narrowest limits no limits inhere. What joy to discern the minute in infinity! The vast to perceive in the small, what divinity!” — Jacob Bernoulli in his Tractatus de Seriebus Infinitis (Treatise on Infinite Series), 1689
A divergent series is a sum such as Eventually as you keep adding these together it just goes off into infinity. (As you might expect, a convergent series converges to some number.)
The Harmonic Series is a special kind of divergent series that looks like this:
The Harmonic series is so named because of the fact that it shows up in music all over the place as overtones, the wavelengths of a vibrating string, each in succession, above the fundamental wavelength.
At first blush, it looks more like a convergent series, but it does turn out to diverge. The quote above is from Jacob Bernoulli, a 17th century mathematician, who, along with his brother Johann and Pietro Mengoli, gave proofs for its divergence.
Sadly, it was ALSO given proof by Nicole d’Oresme, a French Philosopher in the middle ages, but no one seemed to care, and his proof was lost.Yet another example of how culture, humanity, and history conspire against progress. Around the year 1349 nearly half of the entire population of Europe succumbed to the black death. All other factors aside, this was enough to arrest the progress of every field, let alone mathematics, for some time.
Oresme’s proof was pretty cool. I won’t go into details, but let me sum it up (get it?).
He grouped the terms together like this:
… where we count as the first group, then put 2 terms in the 2nd group, 4 terms in the third group, etc. such that each successive group contains terms.
Notice that each group adds up to greater than . Because we have infinitely many groups, we’re “really” just adding numbers bigger than over and over, so this series must diverge off into the infinite.
We’re aren’t supposed to write it this way:
Then again, Euler did it… (Philosophical inclinations play a part in mathematics as they do everywhere else!)
Now go lift something heavy,