Euler's Estimate of Pi

In [1] Euler derived the formula . He claimed that his formula was better for calculation than the Leibniz–Gregory formula , since for , the factor in the series has values , which are simpler to calculate with. He illustrated this with the formula . He calculated eight terms of the sum for each of the arc tangents on the right to 27 decimal places each and concluded that . On the next page he calculated terms 9–16 of the first part and terms 9–10 of the second part and concluded that . To eighteen places, . To 30 places, .


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See [2] for a history of Euler's attempts to calculate an approximation to . The calculations from [1, p. 141] are added.
[1] L. Euler, "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae," Nova Acta Academiae Scientarum Imperialis Petropolitinae 11, 1798, pp. 133-149. www.math.dartmouth.edu/~euler/tour/tour_08.html.
[2] E. Sandifer. "How Euler Did It: Estimating ." MAA Online. Feb 2009. (Jun 20, 2013).
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