Śleszyński–Pringsheim theorem

Template:Short description In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.

It states that if $a_n$, $b_n$, for $n=1,2,3,\dots$ are real numbers and $|b_n|\ge |a_n|+1$ for all $n$, then

$\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\ddots }}}$

converges absolutely to a number $f$ satisfying $0<|f|<1$, meaning that the series

$f=\sum _n\{\frac{A_n}{B_n}-\frac{A_{n-1}}{B_{n-1}}\},$

where $A_n/B_n$ are the convergents of the continued fraction, converges absolutely.

See also

• Convergence problem

Notes and references

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