# Śleszyński–Pringsheim theorem

Jump to navigation
Jump to search

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 *a*_{n}, *b*_{n}, for *n* = 1, 2, 3, ... are real numbers and |*b*_{n}| ≥ |*a*_{n}| + 1 for all *n*, then

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

where *A*_{n} / *B*_{n} are the convergents of the continued fraction, converges absolutely.

## See also

## Notes and references

- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ 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 }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}