Śleszyński–Pringsheim Theorem
   HOME

TheInfoList



Click Here for Items Related To - Śleszyński–Pringsheim Theorem
OR:
ÅšleszyÅ„ski–Pringsheim Theorem on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, the Śleszyński–Pringsheim theorem is a statement about
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
of certain
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s. It was discovered by
Ivan Śleszyński Ivan () is a Slavic male given name, connected with the variant of the Greek name (English: John) from Hebrew meaning 'God is gracious'. It is associated worldwide with Slavic countries. The earliest person known to bear the name was the ...
and
Alfred Pringsheim Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was the father-in-law of the author and Nobel Prize winner Thomas Mann. Family and academic career Pringsheim was born in Ohlau, Prov ...
in the late 19th century. It states that if a_n, b_n, for n=1,2,3,\ldots are
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s and , b_n, \geq , a_n, +1 for all n, then : \cfrac converges absolutely to a number f satisfying 0<, f, <1, meaning that the series : f = \sum_n \left\, where A_n / B_n are the convergents of the continued fraction, converges absolutely.


See also

*
Convergence problem In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ''a'i'' and partial denominators ''b'i'' that are sufficient to guarantee the convergence of the infinite con ...


Notes and references

Continued fractions Theorems in real analysis {{mathanalysis-stub