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Pi, Euler numbers, and asymptotic expansions. (English) Zbl 0711.11009

Intriguing asymptotic expansions are given for the remainders obtained by truncating Gregory’s series for \(\pi\), the alternating harmonic series for \(\log 2\), and the sum of the reciprocals of the squares. For example, for Gregory’s series it is shown that \[ 4\,\sum^{\infty}_{k=n}(-1)^k/(2k+1) = (-1)^n\sum^M_{k=0}2E_{2k}/(2n)^{2k+1}+R_1(M), \] where \(| R_1(M)| \le 2 \,| E_{2M}| /(2n)^{2M+1}\) and the \(E_k\) are Euler numbers defined by \(1/\cosh t=\sum^{\infty}_{k=0}E_ kt^k/k!\). By judicious choices of \(n\) these asymptotic expansions can give more accuracy than is suggested by the usual error term in Taylor series.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B83 Special sequences and polynomials
40A05 Convergence and divergence of series and sequences
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