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- W3036532329 abstract "In [1], Dalzell proved that $pi = frac{{22}}{7} - int_0^1 {frac{{{t^4}{{(1 - t)}^4}}}{{1 + {t^2}}}}$ . He then used this equation to derive a new series converging to π. In [2], Backhouse studied the general case of integrals of the form $int_0^1 {frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}dt}$ and derived conditions on m and n so that they could be used to evaluate π. As a sequel, he derived accurate rational approximations of π. This work was extended in [3] where new rational approximations of π are obtained. Some related integrals of the forms $int_0^1 {frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}P(t),dt}$ and $int_0^1 {frac{{{t^m}{{(1 - t)}^m}}}{{sqrt {1 - {t^2}} }}P(t)dt}$ with P ( t ) being of polynomial form are also investigated. In [4] the author gives more new approximations and new series for the case m = n = 4 k . In [5] new series for π are obtained with the integral $int_0^a {frac{{{t^{12m}}{{(a - t)}^{12m}}}}{{1 + {t^2}}}dt}$ where $a = 2 - sqrt 3$ . The general problem of improving the convergence speed of the arctan series by transformation of the argument has also been considered in [6, 7]. In the present work the author considers an alternative form for the denominators in integrals. As a result, new series are obtained for multiples of π by some algebraic numbers." @default.
- W3036532329 created "2020-06-25" @default.
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- W3036532329 date "2020-06-18" @default.
- W3036532329 modified "2023-09-24" @default.
- W3036532329 title "New pancake series for π" @default.
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- W3036532329 doi "https://doi.org/10.1017/mag.2020.53" @default.
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