Śleszyński–Pringsheim theorem

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

  1. ↑ Слешинскій, И. В. (1889). "Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей". Матем. сб. (in Russian). 14 (3): 436–438.
  2. ↑ Pringsheim, A. (1898). "Ueber die Convergenz unendlicher Kettenbrüche". Münch. Ber. (in German). 28: 295–324. JFM 29.0178.02.
  3. ↑ W.J.Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see Thron, W. J. (1992). "Should the Pringsheim criterion be renamed the Śleszyński criterion?". Comm. Anal. Theory Contin. Fractions. 1: 13–20. MR 1192192.
  4. ↑ Lorentzen, L.; Waadeland, H. (2008). Continued Fractions: Convergence theory. Atlantic Press. p. 129.


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