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Peter
Posts:
3
Registered:
12/6/04


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


I'd certainly appreciate some help with this question. Thanks in advance, Peter _______________________________________________________
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 halfopen 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?)



