Constructing a Subsequence of (Exp(in))n∈N Converging towards Exp(iα) for a Given α∈R - Open Access Library Journal

OALibJ > Vol.2 No.12, December 2015

Constructing a Subsequence of (Exp(in))_n∈N Converging towards Exp(iα) for a Given α∈R ()

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DOI: 10.4236/oalib.1102135 721 Downloads 1,201 Views

Author(s)

Vito Lampret

Affiliation(s)

University of Ljubljana, Ljubljana, Slovenia.

ABSTRACT

For a given positive irrational

and a real t ∈ [0,1), the explicit construction of a sequence

of positive integers, such that the sequence of fractional parts of products

converges towards t, is given. Moreover, a constructive and quantitative demonstration of the well known fact, that the ranges of the functions cos and sin are dense in the interval [-1,1], is presented. More precisely, for any α ∈ R, a sequence

of positive integers is constructed explicitly in such a way that the estimate

holds true for any j ∈ N. The technique used in the paper can give more general results, e.g. by replacing sine or cosine with continuous function f: R→R having an irrational period.

KEYWORDS

Convergence, Dense, Estimate, Exponential, Fractional Part, Integer Part, Irrational, Limit Point, Sequence

Share and Cite:

Lampret, V. (2015) Constructing a Subsequence of (Exp(in))_n∈N Converging towards Exp(iα) for a Given α∈R. Open Access Library Journal, 2, 1-9. doi: 10.4236/oalib.1102135.

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