ISSN:
1432-1785
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract If ξ∈ (0,1) and A=an, nɛℕ is a sequence of real numbers define Sn(ξ,A)∶=Σ{ak∶:k=[nξ]+1 to n}, nɛℕ, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence Sn (ξ,A), nɛℕ, for all ξ in an appropriate set K of real numbers, that the sequence an, nɛℕ, converges to zero. It was shown that such a conclusion is possible if K={ξ,1−ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper: 1) does the convergence of Sn (ξ,A), nɛℕ, for a single irrational number ξ imply an→0. 2) does the convergence of Sn(ξ,A), nɛℕ, for finitely many rational numbers ξ∈ (0, 1) imply an→0. 3) does the convergence of Sn (ξ,A), nɛℕ, for all rational numbers ξ∈ (0,1) imply an→0?
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01170310
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