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On the Convergence of a Class of Nearly Alternating Series

  Published:2007-02-01
 Printed: Feb 2007
  • J. H. Foster
  • Monika Serbinowska
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Abstract

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 fractions Series, convergence, almost alternating, convex, continued fractions
MSC Classifications: 40A05, 11A55, 11B83 show english descriptions Convergence and divergence of series and sequences
Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
Special sequences and polynomials
40A05 - Convergence and divergence of series and sequences
11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
11B83 - Special sequences and polynomials
 

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