That’s what I secretly wished. If it where infinite then everything is explained.

But the Wikipedia article “said” that it is 0 (because 1^0+2^0+3^0+...=1+1+1, and 0 is even). But the [dead?!] article submitted by ColinWright from Tao’s blog says it’s -1/2. (I prefer not to disagree with Tao, just fixed Wikipedia.)

The problem is more deep. I should read the complete version of Tao’s article.

> The problem is more deep.

You want to focus on the idea of analytic continuation. That's where the magic happens.

the claim in the operations is that these sums have finite values.

Yes, but you cannot assume that the values are finite in the "proof" that they are finite. You have to prove that they are finite before you are allowed to manipulate them as finite quantities.

to prove by contradiction, that's exactly what you do. You assume they are finite, use allowed operations, show a contradiction.