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.W. J. Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see
It states that if is a positive integer and , are sequences of real numbers such that for all , then
converges absolutely to a number satisfying , meaning that the series
where are the convergents of the continued fraction, converges absolutely.
Proof Recall that the th convergents of , which will be denoted by in this article, can be computed from the following recurrence relation:
where , , , and . See this article for more detail.
th convergent as a series
First, we will prove the following claim via mathematical induction
By dividing both sides of the claim by , the equation becomes
Thus,
Absolute value of th convergent as a series
Now suppose that for all . Using the recurrence relation of , note that
Thus,
Since by assumption, then by using mathematical induction, one can show that
Consequently, the sequence of is monotone nondecreasing and bounded from below by . Moreover,
Since the right-hand side forms a telescoping series, it is easy to see that
for all values of . Furthermore, the nondecreasing property of the sequence also implies the nondecreasing property of the sequence . Thus, the sequence must converge, by the monotone convergence theorem.
Note that the left-hand side is the upper bound of series representation of after applying triangle inequality, which completes the proof.
See also Convergence problem
Notes and references