Borwein, J. M.; Borwein, P. B.; Dilcher, K. Pi, Euler numbers, and asymptotic expansions. (English) Zbl 0711.11009 Am. Math. Mon. 96, No. 8, 681-687 (1989). 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. Reviewer: Tom M. Apostol (Pasadena) Cited in 17 Documents MSC: 11B68 Bernoulli and Euler numbers and polynomials 11B83 Special sequences and polynomials 40A05 Convergence and divergence of series and sequences Keywords:asymptotic expansions; Gregory’s series for \(\pi \); Euler numbers PDFBibTeX XMLCite \textit{J. M. Borwein} et al., Am. Math. Mon. 96, No. 8, 681--687 (1989; Zbl 0711.11009) Full Text: DOI Link Online Encyclopedia of Integer Sequences: Euler (or secant or ”Zig”) numbers: e.g.f. (even powers only) sec(x) = 1/cos(x). Decimal expansion of the natural logarithm of 2. Decimal expansion of Pi/4. Decimal expansion of 4*Sum_{k=1..500000} (-1)^(k-1)/(2k-1). Decimal expansion of Pi/3. Expansion of e.g.f. tan(x)*sin(x)/2 (even powers only). Decimal expansion of Pi/(2*sqrt(2)). Decimal expansion of (log(1+sqrt(2))+Pi/2)/(2*sqrt(2)) = Sum_{k>=0} (-1)^k/(4*k+1). Decimal expansion of (Pi/2 - log(1+sqrt(2)))/(2*sqrt(2)) = Sum_{k>=0} (-1)^k/(4k+3). Decimal expansion of Sum_{k=1..50000} (-1)^(k-1)/(2k-1). Decimal expansion of 4*Sum_{k=1..50000} (-1)^(k-1)/(2k-1). Decimal expansion of Sum_{k=1..500000} (-1)^(k-1)/(2k-1). Decimal expansion of 2*Sum_{k=1..500000} (-1)^(k-1)/(2k-1). Decimal expansion of Sum_{k=1..5000000} (-1)^(k-1)/(2k-1). Decimal expansion of 2*Sum_{k=1..5000000} (-1)^(k-1)/(2k-1). Decimal expansion of 4*Sum_{k=1..5000000} (-1)^(k-1)/(2k-1). Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only). Expansion of e.g.f. (1/6!)*sin^6(x)/cos(x) (coefficients of even powers only).