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Harmonic Sums and Divergence of Harmonic Series (visual proof)

This is a short, animated visual proof showing a formula for the sum of harmonic sums using triangular diagrams. As a bonus feature (watch until the end), we utilize this formula along with a theorem due to Ernesto Cesáro to prove that the Harmonic Series diverges. #math #manim #harmonicseries #harmonicsum #divergent #diverges #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof If you liked this video, please consider subscribing to the channel and checking out my other videos. For a static version of the visual proof about Harmonic sums, see Roger Nelsen's "Proof Without Words II: More Exercises in Visual Thinking" compendium (page 116) : https://bookstore.ams.org/clrm-14/. To see the result about the divergence of the Harmonic series, check out this paper by Steven J. Kifowit https://stevekifowit.com/pubs/harm2.pdf (proof 39). To learn more about animating with manim, check out: https://manim.community ___________________________________ Music in this video: Cold Blue by Astron