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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
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