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

$$ 
\sum _ {k = 1 } ^  \infty   a _ {k}  =  A \ \ 
\textrm{ and } \ \ 
\sum _ {k = 1 } ^  \infty   b _ {k}  =  B
$$

be convergent series and let the limit

$$ 
\lim\limits _ {n \rightarrow \infty }  
\frac{a _ {n} }{b _ {n} }
  =  \gamma  \neq  0
$$

exist. Then

$$ 
\sum _ {k = 1 } ^  \infty   a _ {k}  =  \gamma B +
\sum _ {k = 1 } ^  \infty  
\left ( 1 - \gamma

\frac{b _ {k} }{a _ {k} }

\right ) a _ {k} .
$$

If the sum  $  B $
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>

====Comments====

====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>