I've been had by Mr. Bender's lectures
Bamboozled, duped, hoodwinked, well, you get the idea. I drank his cool aid on (lack of) mathematical rigor and I bought his idea of summing infinite series like
\( 1+ 2+ 3+ \cdots = -1/12\)
What I did wrong was swallowing hook, line, and sinker his postulates:
- Rule 1: Summation property:
- \(S(a_0 + a_1 + a_2 + \cdots )= a_0 + S(a_1 + a_2 + \cdots )\)
- Rule 2: Linearity:
- \(S(\sum (\alpha a_n + \beta b_n) )= \alpha S(\sum (a_n)) + \beta S(\sum (b_n))\)
Why? Because in addition to assuming a unique result, they are inconsistent. If:
\(1+2+3+ \cdots = -1/12 = A\)
Then for example consider this:
\(A-A = (1+2+3+ \cdots ) - (1+2+3+ \cdots )\)
which by rule 1:
\( 0 = (1+2+3+ \cdots ) - (0+1+2+3+ \cdots ) \)
\(0 = 1+1+1+1+ \cdots\)
Aha! This now implies that
\(0=1 + (1+1+1+\cdots ) = 1+0\) and so \(0 = 1\)!!!!
The two rules work for alternating sums, but when the sign of the sum terms is the same the two rules are clearly inconsistent.
But does this mean that \(1+2+3+ \cdots \) is not \(-1/12\) ? Not at all. The result is still valid due to deeper reasons: analytic continuation of Riemann zeta function.
It is not easy to find why things like this work in math, but in general physics intuition is a very good clue that there must be a solid and rigorous foundation. It is just that physicists' focus is on solving the practical problems and not on the deeper mathematical theory. One may say that a physicist to a mathematician is like an engineer to a physicist :) This is not that bad though: the engineers make more money than physicists, and physicists make more money than mathematicians.
I still regard Mr. Bender's lectures as outstanding, but I should have trusted my mathematical intuition more and not disregarded the alarm bells of mathematical rigor. The inconsistent argument above is due to David Joyce and I ran across it on Quora where the -1/12 result is discussed often.