one of the most beautiful identities i've ever figured out on my own.. (φ is the Golden Ratio)pic.twitter.com/nfRGWISqTl
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i figured this out when i was an undergrad and thought i found something new and then when i showed my professor he said "ah, this was one of my favorites as an undergrad as well" the end
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Assuming convergence
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1/φ <1 so convergence isn't a question
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That's beautiful. My own phi "discovery" (I'm sure I'm not the first) back in college was taking the φ - 1 = 1/φ relationship, multiplying by phi to get φ² - φ = 1 Starting with φ² = 1 + φ you can substitute in the next power: φ³ = φ(1 + φ) = 1 + 2φ (continued)
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... and continuing on: φ² = 1 + φ φ³ = 1 + 2φ φ⁴ = 2 + 3φ φ⁵ = 3 + 5φ I was pleased to see the Fibonacci sequence appear. Negative powers of phi gets you a similar sequence.
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So, this would work for any constant greater than one, like e and pi, but not gamma. Euler's going to be pissed when you tell him...
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well no, because you need the φ recursion relation
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Here is a direct proof that also shows that the solution is unique (by using a condition for convergence of the geometric series).pic.twitter.com/kWCtOCoyIw
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If you like this, watch this video (and especially the follow-up video with the proof).https://youtu.be/FtNWzlfEQgY
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Now that was cool.
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