location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword continued fractions

 Expand all        Collapse all Results 1 - 2 of 2

1. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
 On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $\alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$. Keywords:Series, convergence, almost alternating, convex, continued fractionsCategories:40A05, 11A55, 11B83

2. CJM 2002 (vol 54 pp. 1305)

Vulakh, L. Ya.
 Continued Fractions Associated with $\SL_3 (\mathbf{Z})$ and Units in Complex Cubic Fields Continued fractions associated with $\GL_3 (\mathbf{Z})$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields. Keywords:fundamental units, continued fractions, diophantine approximation, symmetric spaceCategories:11R27, 11J70, 11J13