Calculate the difference between π/4 and the Leibniz series for computing &#x03C0;<!-- π -->

Raul Walker 2022-07-08 Answered
Calculate the difference between π/4 and the Leibniz series for computing π / 4 with n = 200.
This series appears to converge relatively slowly, and so at what point can we confidently say that "the 576th digit is 3" of an irrational number?
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Answers (1)

postojahob
Answered 2022-07-09 Author has 13 answers
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 R k = n = k + 1 a n ). Say you want m digits, then you want 10 m > | R k | .
For the Liebniz series n = 0 ( 1 ) n 2 n + 1 , since it is an alternating series, we know that the remainder term is bounded by 1 2 n + 1 itself, and so if you want m digits of π 4 , 10 m 1 2 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.
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