Is quite easy: let S be the total sum of the series. If you divide by 2 every term you get 1/2+1/4+1/8 etc, i.e. S - 1. So
S/2 = S - 1 - > S/2 = 1 - > S = 2
S/2 = 1
This can be generalized for every series of 1/n^m. It can be demonstrated that the sum converge if and only if m is > 1. The first series ha m = 1 and does not converge.
I get the first one now. By definition every nth term will be halfway between the last point and 2, so the series will never exceed 2. With infinite terms it becomes 2.
We’ve been over that before. Looking at it as a Dirichlet series, and then extending the zeta function by its analytic continuation, it “equals” that, yeah, sorta. This explains the whole story: https://plus.maths.org/content/infinity-or-just-112
Yeah, soon after I posted that I found a video explaining how the usual demonstration had been debunked. It concluded that it had still some validity but using “equals” would be an exaggeration.