Let K(a(n)/b(n)) be a continued fraction with elements (a(n), b(n)) picked randomly and independently from C-2 according to some given probability distribution. Then K(a(n)/b(n)) converges with probability 1 under mild conditions on the measure. We extend this to random iterations of Mobius transformations and to the p-periodic case where {(a(np+k), b(np+k))}(n=1)(infinity) is picked randomly and independently from a distribution depending on k for k = 1, 2,..., p.

Let K(an/bn) be a continued fraction with elements (an,bn) picked randomly and independently from (C∖{0})×C according to some probability distribution μ. We show that K(an/bn) converges generally with probability 1 under very mild conditions on the measure μ.

Let K(a(n)/b(n)) be a continued fraction with elements (a(n), b(n)) picked randomly and independently from (C \ {0}) x C according to some probability distribution mu. We find sufficient conditions on mu for K(a(n)/b(n)) to converge generally with probability 1.

Let K(a n /b n ) be a continued fraction with elements (a n ,b n ) picked randomly and independently from \((\mathbb{C}\setminus\{0\})\times\mathbb{C}\) according to some probability distribution μ. We find sufficient conditions on μ for K(a n /b n ) to converge with probability 1 or to be restrained with probability 1. More generally, we also consider μ-random sequences {τ n } of independent Möbius transformations and find sufficient conditions for \(\{\tau_{1}\circ\tau_{2}\circ\cdots\circ\tau_{n}\}_{n=1}^{\infty}\) to converge or be restrained with probability 1. The analysis is based on an important paper by Furstenberg.

The purpose of this paper is to tell how continued fraction expansions of functions are derived, how they relate to Padé approximation, and how they can improve Padé approximants.

We prove that the Ramanujan AGM fraction diverges if |a|=|b| with a 2≠b 2. Thereby we prove two conjectures posed by J. Borwein and R. Crandall. We also demonstrate a method for accelerating the convergence of this continued fraction when it converges.

Periodic continued fractions are well understood. We can determine when they converge, when they diverge, their values if they converge and the asymptotic behavior of their tail sequences. Indeed, it is all a matter of iterations of linear fractional transformations, and we have closed expressions for their approximants Sn(w).

The convergence theory for continued fractions relies on some basic concepts and ideas. The first part of this chapter is devoted to some of these tools. Next we go on to present some classical convergence theorems. They are classical in two meanings of the word. For one thing they are old, well-proven results. But they are also classical in the sense that they aim for classical convergence. The proofs we o®er are not always the classical ones, though. We have chosen to see the theorems in a more modern setting whenever convenient.

We prove that if the continued fraction K(a(n)/1) has circular twin value sets < V-0, V-1 >, then K(a(n)/1) converges except in some very special cases. The results generalize previous work by Jones and Thron.

If K(an/bn) converges generally to f, then its approximants {Sn(wn)} converge to f as long as {wn} stays asymptotically away from its exceptional sequence. Therefore we can use Sn(wn) as an approximation to f. However, the choice of {w makes quite a difference. For one thing, if one is looking for, say, a rational approximant, then this limits the choice for {wn. But often one just wants fast convergence to f. In this chapter we suggest a number of ideas for how to choose wn to this aim.

We have often been asked questions, by students as well as by established mathematicians, about continued fractions: what they are and what they can be used for. Sometimes the questions have been raised under circumstances where a quick answer is the only alternative to no answer: in the discussion after a talk or lecture, by a cup of co®ee in a short break, in an airplane cabin or on a mountain hike. In responding to these questions we have often been pleased by the sparks of interest we have seen, indicating that we had managed to transmit a glimpse of new and apparently appealing knowledge. In quite a few cases this led to further contact and 'follow-up activities'.

We present an idea on how Ramanujan found some of his beautiful continued fraction identities. Or more to the point: why he chose the ones he wrote down among all possible identities.

We consider linear fractional transformations T n which map the unit disk U into itself with the property that T_n(U)⊆ T_n-1(U)⊆ U for all n . Clearly, the closed sets T_n(U) form a nested sequence of circular disks, and thus has a non-empty limit set T_∞ (U) . If this limit set is a single point, then T n (w) converges uniformly in U to this point. In this paper we study what happens if the limit set has a positive radius. In particular we prove that under specific conditions, the derivatives satisfy ∑ |T_n'(w)|<∞ for w∈ U and T n (w) still converges locally uniformly in U to a constant function. Results of this type are useful in the theories of dynamical systems and continued fractions.

In the late 1990s failure rates in a first-year introductory calculus course at the Norwegian University of Science and Technology reached peak levels. This paper reports on findings from an action research project that was set up in 2002/2003 to improve the situation. The study confirms that students approach their tasks differently which contributes to qualitatively different learning outcomes. Furthermore, patterns of achievement in mathematics and physics in secondary education keep reoccurring in the calculus course, even though the teaching and learning contexts are different. The paper does not provide any definite answer as to why groups of students get involved in distinctly different learning processes, and it will take further research to decide the nature of commitment to the learning tasks. However, inspired by the notion of ‘practices’ this paper raises a discussion about the role of intentionality in learning processes. When doing mathematics, students are also in a process of being engaged in and developing a practice. It is a major challenge for academic staff to contribute to communities of practice that are conducive to learning.

Modified approximants of a continued fraction are designed to increase the rate of convergence, and these led to the notion of restrained sequences of Mobius transformations. Here we give some analytic and geometric characterizations of restrained sequences and related topics. We also give an expository account of the use of geometry, including hyperbolic geometry, in discussing restrained sequences and continued fractions.

Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.