I Probability fun time: Proof that 1/3=1/2=1/4

Frabjous
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Forgive me, I am not a probability guy, so I am unsure how well known this is. I was trying to figure something out and found this. I found it cool.

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Here's the explanation.

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The first solution is a fraction (damn scanner!)

Oops! From Kendall Geometrical Probability (1963)
 
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The basic idea is reasonably well-known. For example, if we have two concentric circles of radii ##1## and ##2## and we pick a point "at random" in the larger circle, what is the probability it is in the smaller circle?

If we choose Cartesian coordinates uniformly, then the probablity is ##1/4##. But, if we choose polar coordinates, then the probability is ##1/2##.
 
When you use polar coordinates, the correct was (equal area) is uniform in angle and uniform in ##r^2## (not in ##r##).
 
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Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

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