I went full @3blue1brown and used Manim to illustrate the proof of the Geometric Series formulapic.twitter.com/c3kVy63SIh
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Here's the Geometric Series proof more rigorously, before someone accuses me of being a physicist — waitpic.twitter.com/R8KTw4rcSS
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Here's my python source code for the animation. The library manimlib is
@3blue1brown's graphics engine.https://github.com/InertialObservr/manim_projects/blob/master/geometric_series.py …Show this threadThanks. Twitter will use this to make your timeline better. UndoUndo
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In my 1st job interview after graduating (i.e. for anything other than flipping burgers or delivering milk) they asked if I knew the (finite) geometric series formula. I couldn’t remember it, so I derived it, because frankly that’s easier than remembering if it’s x^n or x^{n+1}.
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very relatable, i'm the exact same way .. the derivation is essentially one "trick" to remember, as opposed to keeping track of all of the ns and n+1 s entering and leaving your life
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BUT WHAT ABOUT WHEN x=1????
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The last term of (1) should read x^(n-1). Then, when you multiply (2) by r, the last term becomes x^n. This is an important detail, b/c to claim the infinite sum converges, x^(n) must vanish as x approaches +infinity. This can only happen when |x|<1, which I'm sure u know
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Ending at n or n-1 doesn’t change the proof at all , line 1 is fine
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There is no need to disallow x=0. Also, this proof doesn't explain why there is any restriction on the value of x, hence does not really establish convergence for |x|<1.
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