I was watching this passage from a course on history of maths, where harmonic series, inverse square series and other ones are described in the context of the more generic zeta function:
zeta function, cases n=1 and n=2
I was looking for some geometrical interpretation for this, and I found out something weird, when comparing cases n=1 and n=2.
For n=1, I assumed I was starting with a segment of length 1, then adding a half, then a third and so on: the "resulting segment" will be infinitely long, as that series (the harmonic one) does not converge.
For n=2, I assumed I was playing the same segment game, but in 2D, adding areas of 1, one fourth, one ninth, and so on: this series (the inverse square one) converges to Pi^2/6.
Question 1: Is such an interpretation consistent?
I mean: it brings (apparently) weird results!
Consider the "resulting segment": at each iteration, it is still getting longer, indeed it gets to infinite length, but each "added square" is getting so smaller so fast that the area is not getting infinite: quite counter intuitive.
Indeed, since at each iteration the "resulting segment" is getting longer, then the "added piece" wouldn't be of zero-length, so why should a square build on that "added piece" would be of zero-area?
Question 2: how would you explain, geometrically, this behaviour?