Recent results relevant to the evaluation of infinite series (Q760619)

This paper deals with the Shanks transformation \(E_ n(c_ r)\) for convergence acceleration of a sequence \(\{c_ r\}\). The type of sequences considered are \(a)\quad c_ r\sim (r+\mu)^{\epsilon}q^ r\sum^{\infty}_{t=0}x_ t(r+\mu)^{-t},\) \(r\to \infty\) where \(x_ 0=1\), \(\mu\geq 0\), \(| q| \leq 1\), \(q\neq 1\), \(\epsilon\in {\mathbb{C}}\); \(b)\quad c_ r\sim (r+\mu)^{e}q^{r+1} \sin \eta_ r,\) \(r\to \infty\) where \(\mu\geq 0\), \(0<q\leq 1\), \(\epsilon\in {\mathbb{C}}\setminus \{0,1,2,...\}\), \(\eta_ r=r\theta +\delta\), \(0<\theta <\pi\), \(\delta\in {\mathbb{R}}\); \(c)\quad c_ r\sim (-1)^ r[(r+\mu)^{\epsilon}+\phi],\) \(r\to \infty\) where \(\phi\) is a constant, \(\mu\geq 0\), \(\epsilon\in {\mathbb{C}}\setminus \{0\}\); d) \(c_ r\sim a_ k/(r+\mu)^ k,\) \(a_ k\neq 0\), \(r\to \infty\) where \(k\geq 1\), \(\mu\geq 0\). The method is actually equivalent to using the Padé approximant at \(z=1\). This paper is a continuation of previous papers of the authors [J. Math. Phys. 19, 821- 829 (1978; Zbl 0387.65003) and ibid. 19, 2405-2409 (1978; Zbl 0438.65003)]; the second author [ibid. 21, 112-119 (1980; Zbl 0448.65003)].