Math Forum Discussions

Peter

Posts: 3
Registered: 12/6/04

Proof of Irrationality
Posted: Aug 13, 2010 9:31 AM

Plain Text

I'd certainly appreciate some help with this question. Thanks in advance, Peter
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Let L(x) be the power series

L(x) = sum_{i = 1}^{00} x^{i !} = x + x^2 + x^6 + x^24 + . . . . ,

and let r be some rational such that 0 < r <= ½.

I think that, for all r in the given half-open interval, L(r) is irrational.
Can this be proved ?
Or maybe I are wrong and there is/are just some value(s) of r for which L(r) is irrational.

P.S. Does the series L(x) have a name at all ? I think it is what is known as a Lacunary function (i.e., it has a so called ?natural boundary?)