Difference between revisions of "Kummer transformation"
From Encyclopedia of Mathematics
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A transformation of series of numbers which improves the convergence; proposed by E. Kummer. Let | 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 | be convergent series and let the limit | ||
| − | + | $$ | |
| + | \lim\limits _ {n \rightarrow \infty } | ||
| + | \frac{a _ {n} }{b _ {n} } | ||
| + | = \gamma \neq 0 | ||
| + | $$ | ||
exist. Then | 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung", '''2''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp, "Infinite sequences and series", Dover, reprint (1956) (Translated from German)</TD></TR></table> | ||
Latest revision as of 11:30, 16 April 2023
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) |
How to Cite This Entry:
Kummer transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_transformation&oldid=17267
Kummer transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_transformation&oldid=17267
This article was adapted from an original article by V.V. Senatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article