View source for Kummer transformation

A transformation of series of numbers which improves the convergence; proposed by E. Kummer. Let

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be convergent series and let the limit

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exist. Then

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If the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056000/k0560004.png" /> is known, the Kummer transformation may prove useful in computations, since the series on the right converges more rapidly than that on the left.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,   "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,   "Infinite sequences and series" , Dover, reprint  (1956)  (Translated from German)</TD></TR></table>