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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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 &quot;the   576th digit is   3&quot; of an irrational number?

postojahob · Accepted Answer

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        &amp;gt;      |                  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&#039;s theorem can be used to provide a bound on the remainder term.

Calculate the difference between π/4 and the Leibniz series for computing π

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