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Remarks concerning a method for accelerating the convergence of sequences. (English) Zbl 0819.65005
The main result may be presented as follows. Denote the differences of a sequence $$a(n)$$ $$(n \geq 0)$$ by $$\Delta (a \| r | n)$$, so that $$\Delta (a \| 0 | n) = a(n)$$, $$\Delta (a \| 1 | n) = a(n + 1) - a(n)$$ and so on. Let $$S(n)$$ $$(n \geq 0)$$ be a monotonic sequence and $$a(n)$$ $$(n \geq 0)$$ be an auxiliary sequence for which, as $$n$$ increases, $$\lim \Delta (a \| p - 1 | n) = L$$ exists and is finite, $$p \geq 1$$ being fixed. Set $$B(a,S \| p,k | n) = \Delta (a \| p | n)/ \Delta (S \| 1 | k + n)$$, $$k \geq 0$$ also being fixed. If, as $$n$$ increases, $$\lim B(a,S \| p,k | n)$$ is finite and nonzero then $$S = \lim S(n)$$ exists and, setting $$T(n) = S(n) + \{L - \Delta (a \| p - 1 | n - k) /B(a,S \| p,k | n)$$, the ratio $$\{S - T(n)\}/ \{S- S(n)\}$$ tends, as $$n$$ increases, to zero.
Reviewer: P.Wynn (Mexico)
##### MSC:
 65B05 Extrapolation to the limit, deferred corrections 40A05 Convergence and divergence of series and sequences