S3E50. Why Zeta(1) diverges but Zeta(2) converges

In this episode we go over three "as simple as possible but not simpler" proofs that show, in that order, that the infinite sums Zeta(1/2) and Zeta(1) diverge, while the infinite sum Zeta(s greater than 1) converges. In order to develop a feel for the local material, we, first, try our hand at something simple, the infinite sum Zeta(1/2). While doing so we develop a miniature framework that will serve us well for the duration of this episode. Start with an evident true relation, such as "k is less than n" and then spin that relation into something useful. Next, we go back more than 600 years and, using the argument that, originally (and in Western Europe) belongs to a French philosopher Nicole Oresme (1325-1382), show that the infinite series Zeta(1) diverges. The cleverness of the Oresme's argument boils down to two tactical maneuvers. One, split the monolithic number n into a sum of smaller numbers. Two, do so on the boundaries of perfect powers of 2. In the process we also explain why the Zeta(1) sum is called "harmonic". Spoiler alert, because starting with n=2 every term of such a sum is a harmonic mean of its two immediately adjacent neighbors, the left and the right. Lastly, playing for all the marbles, we craft an argument that shows that for any (real) number s strictly larger than 1 the corresponding harmonic series Zeta(s) converges. As an extra for experts, we take a brief look at the miracle of the said zeta functions of an even argument, infinite sums that admit the existence of a finite description of the real numbers that these sums converge to. Before looking at the next challenge, in physics, we also mention a practical application of the material covered in this episode, the Raabe's convergence test, because a popular proof of the fact that this test does work (in certain cases) plows directly through and relies on the properties of the "general harmonic" series Zeta(s) that we discussed.