Śleszyński–Pringsheim theorem

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In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński[1] and Alfred Pringsheim[2] in the late 19th century.[3]

It states that if an, bn, for n = 1, 2, 3, ... are real numbers and |bn| ≥ |an| + 1 for all n, then

converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1,[4] meaning that the series

where An / Bn are the convergents of the continued fraction, converges absolutely.

See also

Notes and references

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  3. W.J.Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  4. {{#invoke:citation/CS1|citation |CitationClass=book }}


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