On Irrational Valued Series

Abstract

There are many rational termed convergent series in analysis that sum to an irrational number. One well-known example can be found via the Taylor expansion of the exponential function, where in particular the base of the natural logarithm is represented as an infinite sum of the reciprocals of n\. The irrationality of e can be deduced directly from this series via an argument of Euler's (see [2]). In recent times, a number of authors [3], [4] have noted that other irrational valued series may be constructed by replacing n\ in the series for e by the product vxv2. ..vw, where {vw} is a strictly monotone increasing sequence of positive integers. However, in such cases, one needed to impose the additional assumption that n\v$>2...vn for each n. In this paper we shall demonstrate that irrational valued series may similarly be constructed from the terms of a generalized Fibonacci sequence, which are generated via the recurrence relation

Cite this paper

@inproceedings{Nyblom1997OnIV, title={On Irrational Valued Series}, author={M. A. Nyblom}, year={1997} }