Edit: mental math botched. I thought I remembered the powers of 2 better than I actually did. 160 characters would be about 10k, and 100k would buy you a whopping 163 characters.
Incorrect, there is actually some well-established mathematics dealing with divergent series as summing to finite values via analytic continuation. These results have to be treated with care, because they are inconsistent under certain algebraic manipulations, but the techniques are "real" enough that their results are used in particle physics.
"The sum of the series 1+2+3+4+5+6... = -1/12" is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series.
Even mathematicians working with Cesàro sums surround such statements with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." [0]
Precisely none of the times I've heard the "1+2+3+4...=-1/12" bullshit has the person stating it prefaced their statement with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..."
If you say that "1+2+3+4...=-1/12" without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that "this 95 cent pack of gum is free" because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum.
The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers.
[0] Madras, Neal. "A Note on Diffusion State Distance." arXiv preprint arXiv:1502.07315 (2015).
Yeah, I botched my mental math, but yours isn't quite right either, because the 141st character is 1 cent, which is 2^0. So the 20th power of two in this context is 2^19, and the sum of the powers of two up to the 19th is 2^20 - 1.
Damn. I had originally even typed it with 2^20-1 and $10k, but then second-guessed myself. (This process of me second guessing my math intuition and then making it worse is also why I didn't end up competing on the national level for my state in the high school math leagues ;P.)
Edit: mental math botched. I thought I remembered the powers of 2 better than I actually did. 160 characters would be about 10k, and 100k would buy you a whopping 163 characters.