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Curated by: DFulks Math (15 videos)
Where does this beautiful series come from (called Leibniz series)? π/4 = 1 − 1/3 + 1/5 − 1/7 + ... In this video, we first derive the series using calculus: geometric series, then arctangent function, and an integral. Later, we go over the basic idea behind a very different approach that uses geometry and algebra, without derivatives, integrals, or the arctangent function. Limits are still involved, of course, since we are dealing with an infinite series. We also discuss a very strange feature of the Leibniz series. Although the series converges extremely slowly, at certain stopping points the approximation to π contains surprisingly long stretches of correct decimal digits. This phenomenon is related to deeper ideas such as Euler–Maclaurin summation. I have also included a PDF below with a more complete explanation of the algebra-and-geometry approach: [PDF LINK HERE] https://drive.google.com/file/d/1FhbhSYCRqsH9XvbvOjeOm3jL1dIXCyBE/view?usp=sharing If you would like to see a full video devoted to the non-calculus derivation, let me know in the comments. Thanks for watching, and please like and subscribe if you would like to see more math videos like this.