Wallis' sequence estimated accurately using an alternating series

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作者：Vito Lampret ^{vito.lampret@guest.arnes.si" class="auth_mail" title="E-mail the corresponding author}
关键词：primary ; 26D20 ; 40A25 ; 41A60 ; secondary ; 11Y60 ; 11Y99 ; 65B15
刊名：Journal of Number Theory
出版年：2017
出版时间：March 2017
年：2017
卷：172
期：Complete
页码：256-269
全文大小：394 K

文摘

An asymptotic approximation of Wallis' sequence ii=S0022314X16302220&_rdoc=1&_issn=0022314X&md5=678da7631bd22d2adb84a0564bb92157">

m \mapsto W_{m} : = \prod_{k = 1}^{m} \frac{4 k^{2}}{4 k^{2} - 1}

is presented as

ii=S0022314X16302220&_rdoc=1&_issn=0022314X&md5=288ca66a48b88c32dd58cb9a425ec5d4">

W_{m} = \frac{m π}{2 m + 1} \exp (2 σ_{q} (m)) \cdot \exp (r_{q} (m)),

where

ii=S0022314X16302220&_rdoc=1&_issn=0022314X&md5=03db939bb19b11294f242df688a50b7f">

\begin{matrix} σ_{q} (x) : = \sum_{i = 1}^{⌊ q / 2 ⌋} \frac{(1 - 4^{- i}) B_{2 i}}{i (2 i - 1) \cdot x^{2 i - 1}} \\ (B_{k} are the Bernoulli coefficients), \end{matrix}

and where

ii=S0022314X16302220&_rdoc=1&_issn=0022314X&md5=19036c6b58ff1732222635ffc0e776a4">

\begin{matrix} | r_{q} (m) | < r_{q}^{⁎} (m) : = \frac{2 π (q - 2)!}{3 {(2 m π)}^{q - 1}}, \\ for any integers m \geq 1 and q \geq 2 . \end{matrix}

Parameters m and q control the error factor ii=S0022314X16302220&_rdoc=1&_issn=0022314X&md5=69ec90a266d3063aad8e2dafb4d958b0" title="Click to view the MathML source">exp⁡(r_q(m))

\exp (r_{q} (m))

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