I am a physicist interested in physics. In particular this question is related to quantum field theory.
I recently came across a derivation of the infinite sum $1+1+1+1+..... $ that produced the result -1/2, aka zeta regularization (from Terry Tao's blog)
This was quite surprising to me as I had previously known an infinite sum of 1s to be divergent - from taking a math physics course by the guy that wrote "the book" on asymptotic methods.
(Indeed I've known the sum of all positive integers to be finite for quite some time as well as many other "divergent-looking" sums and I get the whole idea behind summation methods like Padé, Shanks, Euler, etc.)
Anyways this prompted me to wonder;
- Are ALL infinite sums not divergent?
- If not then how can one determine whether a sum os divergent or not?
- What was all this business in undergrad calculus about learning tests of convergence and all this business in complex analysis about series if weird things like $1+1+1+1+.....$ are actually convergent??
I'm still confused by all this stuff. And I haven't found an answer to these questions that "click"
Any help understand this topic would be kindly appreciated.