Calculate the difference between π/4 and the Leibniz series for computing
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Raul Walker 2022-07-08Answered
Calculate the difference between π/4 and the Leibniz series for computing with . This series appears to converge relatively slowly, and so at what point can we confidently say that "the th digit is " of an irrational number?
The way you can tell how many digits you have computed is by providing a bound on the remainder term for the series (i.e., of ). Say you want m digits, then you want . For the Liebniz series , since it is an alternating series, we know that the remainder term is bounded by itself, and so if you want digits of , terms would be sufficient (quite a few). For other series, sometimes Tayor's theorem can be used to provide a bound on the remainder term.