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by Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui and Brendan M. Quine

Mathematics 2021, 9(17), 2162; https://doi.org/10.3390/math9172162 - 4 Sep 2021

Cited by 2 | Viewed by 3042

In our earlier publication we have shown how to compute by iteration a rational number

u_{2, k}

in the two-term Machin-like formula for

π

of the kind [...] Read more.

In our earlier publication we have shown how to compute by iteration a rational number

u_{2, k}

in the two-term Machin-like formula for

π

of the kind

\frac{π}{4} = 2^{k - 1} arctan (\frac{1}{u_{1, k}}) + arctan (\frac{1}{u_{2, k}}),

k \in Z, k \geq 1,

where

u_{1, k}

can be chosen as an integer

u_{1, k} = ⌊a_{k} / \sqrt{2 - a_{k - 1}}⌋

with nested radicals defined as

a_{k} = \sqrt{2 + a_{k - 1}}

and

a_{0} = 0

. In this work, we report an alternative method for determination of the integer

u_{1, k}

. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set

\{a_{k}\}

in computation of the integer

u_{1, k}

. Mathematica programs validating these results are presented. Full article

(This article belongs to the Section E: Applied Mathematics)

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