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I'm having "Lebesgue's Integral and Fourier Series" as subject this year in college, and my professor not only marks us by our work, but also by our attitude, or even our capability to negotiate our own grade. The thing is I'm doing pretty well, but I noticed last time there was something that helped improve the grade a little bit: I found over the internet an article by Desmond MacHale about 13 (general) questions you can ask in a mathematical conference which you didn't understand at all. It made him laugh, like a lot. He liked it so much that he even showed it to the rest of professors, and he read it out loud in his classes. There are a couple of works left to do this course, and I believe he is expecting more material of this type. Can anyone recommend me some of it? It hasn't got to be funny, it could be sometwhat entertaining in an interesting or curious way. Thanks in advance.

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Actually, I lied, I can help: all of this is blatantly stolen from Appendix E of Samir Siksek's excellent Introduction to Abstract Algebra notes (which is inexplicably missing from the version that I can find online - this is a new thing, it was definitely there last time I went looking). This appendix is entitled The Forgotten Joys of Analytic Irresponsibility and opens with a disclaimer ending with

Needless to say the material in this appendix is merely for the mortification of fellow Warwick academics. Students must stop reading at once or they’ll do irreparable damage to their souls.

There follow two wonderful things which I shall rather generously call proofs, of which the second (apparently taken from MathOverflow, though I've never found it there) reads:

Let $\int = \int_0^x$. We wish to solve the integral equation $$f(x) - \int_0^x f = 1.$$ Factoring out the $f$, we have $$\left(1 -\int\right)f = 1, $$ and hence \begin{align*}f &= \left(1 - \int\right)^{-1}1\\&= \left(1 + \int + \int \int + \int \int \int + \cdots\right)1\\ &= 1 + \int_0^x1+\int_0^x\int_0^x1+\int_0^x\int_0^x\int_0^x1+\cdots\\&= 1 + x + \frac{x^2}{2!} + \frac{x^2}{3!}+\cdots\\&=e^x.\end{align*}

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  • $\begingroup$ This feels like a fancy analog to anomalous cancellation. Except, It seems like more than just a coincidence. I wonder if there is any meaningful explanation for why this "works out"? $\endgroup$
    – monguin
    Feb 26, 2022 at 3:17
  • $\begingroup$ Sure, $\int$ as defined is the Volterra operator, whose boundedness yields the applicability of the Neumann series expansion to $(I - \int)^{-1}$. It is just this expansion that is used here. $\endgroup$
    – akkapi
    Mar 11, 2022 at 15:09

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