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A modification of the Euler-Abel transform for convergent series. (English) Zbl 0723.40001

Let (1) \(\sum^{\infty}_{k=0}(-1)^ ka_ kx^ k\) \((x>0;a_ k>0\), \(k=0,1,2,...)\) be a given power series with radius of convergence \(\rho (0<\rho \leq +\infty)\). It is well known that its Euler-Abel transform \[ (2)\quad (1/(1+x))\sum^{\infty}_{p=0}(-1)^ p\Delta^ p(a_ 0)(x/(1+x))^ p \] (where \(\Delta^ p(a_ 0)\) represents the pth finite difference of \(a_ 0)\) converges if the original series (1) does. In general, however, nothing can be concluded about the relative rates of convergence since there are examples for which the transformed series (2) converges faster than the original series and vice versa.
Our goal is to obtain a modification of (2) which transforms (1) into an equivalent convergent series and to give a condition which guarantees that the modified series converges faster than (1).

MSC:

40A05 Convergence and divergence of series and sequences
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