Kummer transformation

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

[1]	G.M. Fichtenholz, "Differential und Integralrechnung", 2 , Deutsch. Verlag Wissenschaft. (1964)
[a1]	K. Knopp, "Infinite sequences and series", Dover, reprint (1956) (Translated from German)