In mathematics, the Śleszyński–Pringsheim theorem is a statement about
convergence of certain
continued fractions. It was discovered by
Ivan Śleszyński and
Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was born in Ohlau, Prussian Silesia (now Oława, Poland) and died in Zürich, Switzerland.
Family and academic career
Pringsheim came ...
in the late 19th century.
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,
meaning that the series
:
where ''A''
''n'' / ''B''
''n'' are the
convergents of the continued fraction,
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
.
See also
*
Convergence problem
Notes and references
Continued fractions
Theorems in real analysis
{{mathanalysis-stub