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A dissection proof of Leibniz’s series for \(\pi /4\). (English) Zbl 1298.41044

Summary: Inspired by Lord Brouncker’s discovery of his series for ln 2 by dissecting the region below the curve \(1/x\), Viggo Brun found a way to partition regions of the unit circle so that their areas correspond to terms of Leibniz’s series for \(\pi /4\). Brun’s argument involved ad hoc methods which were difficult to find. We develop a method based on usual techniques in calculus that leads to Brun’s result and that applies generally to other related series.

MSC:

41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
26A06 One-variable calculus
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