Geodesic
On the Convergence of a Class of Nearly Alternating Series
Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 85-108

Voir la notice de l'article provenant de la source Cambridge University Press

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 $\left( {{c}_{k}} \right)\,\in C$ , then $\sum{{{(-1)}^{\left\lfloor k\alpha\right\rfloor }}}{{c}_{k}}$ converges if and only if ${{c}_{k}}\log k\to 0$ .If $\alpha$ is of the second type and $\left( {{c}_{k}} \right)\,\in C$ , then $\sum{{{(-1)}^{\left\lfloor k\alpha\right\rfloor }}}{{c}_{k}}$ 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 $S\left( n \right)\,=\,{{\sum\nolimits_{k=1}^{n}{\left( -1 \right)}}^{\left\lfloor k\alpha\right\rfloor }}$ and double partial sums $T\left( n \right)\,=\,\sum\nolimits_{k=1}^{n}{\,S\left( k \right)}$ .
DOI : 10.4153/CJM-2007-004-1
Mots-clés : 40A05, 11A55, 11B83, Series, convergence, almost alternating, convex, continued fractions 
Foster, J. H.; Serbinowska, Monika. On the Convergence of a Class of Nearly Alternating Series. Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 85-108. doi: 10.4153/CJM-2007-004-1
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