In your step (1) don't take limit just yet! Equation (1) holds true for all n (without the lim). Divide both sides by x and, instead of using Taylor series, use fact that sin(y) / y goes to 1 as y goes to 0, where y = x/2^n. That would then prove (2) rigorously.
Yesterday @AnalysisFact posted Viete's mysterious infinite product that converges to π, so I thought I'd follow up with the proof.pic.twitter.com/iFQlI8PM2n
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But then I wouldn’t have had the opportunity to say “Eulerian bad boy”
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How to show it geometrically?
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You should ask
@3blue1brown about that one.. - 1 more reply
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I lost it at “...I am an eulerian bad boy”
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Yeah if I had come up with it it would have been my Twitter bio
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Show off.
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Mark Kac has a great book using this as a jumping off point for how probablistic notions get into analysis: http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf …
End of conversation
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