“Fun With Math: How To Make A Divergent Infinite Series Converge” is an interesting article about how to make the harmonic series converge by systematically discarding terms. Shockingly, the article appeared in Forbes.
‘I was having dinner with a visiting colleague this week when talk turned to what we were teaching this term. He mentioned the part of calculus dealing with infinite series (the bane of many students) and how really he just mentally compares everything with the harmonic series: 1 + (1/2) + (1/3) + (1/4) + … This series diverges; that is, the sum is infinite (contrast this with the convergent series (1/2) + (1/4) + (1/8) + … = 1). I then casually mentioned that if you take the harmonic series and throw out the terms whose denominators contain a 9 then the resulting series converges.
“I don’t believe that. How can that be true?”
That was my first reaction to hearing this fact, too, because we get so used to the idea that the harmonic series diverges that we can’t believe that throwing out a few terms, even infinitely many, will make a difference. And, there’s nothing special about 9; you can toss out terms containing any particular digit. In fact, you can pick any finite string of digits, toss out the terms containing those, and the result converges. With that set-up, let’s talk about what all this means and how we can prove it.‘
H/T Geek Press.
In the article, the author provides a proof of the divergence of the harmonic series. I showed a much simpler proof using the Cauchy Condensation Test.