On certain transformations of series. (English. Russian original) Zbl 0702.40002

Sov. Math. 32, No. 4, 120-122 (1988); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1988, No. 4 (311), 82-84 (1988).
Let \(\sum^{\infty}_{n=1}a_ n\) be a series of real or complex numbers and let \(r\neq 1\) be a real or complex number. Then the following formula is well-known: \[ (1)\quad \sum^{\infty}_{n=1}a_ n=\frac{a_ 1}{1-r}+\frac{1}{1-r}\sum^{\infty}_{n=1}(a_{n+1}-ra_ n). \] The authors give some generalizations of this formula and using these generalizations they derive some results on accelerating the convergence of infinite series.
Reviewer: T.Šalát


40A05 Convergence and divergence of series and sequences