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Re: subseries of harmonic series converges iff ...
Posted:
Nov 21, 2004 9:11 AM
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Yesterday, The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote,
in part:
> > >I recall reading a theorem once that says
> > > For a subset S of Z+ , the following are equivalent:
> > > * \Sum_{s \in S} 1/s < \infty
> > > * Something else.
> > >
> > >Does anyone know what that something else is?
> >
> > I was hoping someone else would give the correct answer but I was
> > under the impression there's a correct "Something else" of the form,
> > * { x^s, s \in S } is a basis for C[0,1]
> > (the set of continuous functions on the interval). Or something like that.
> This is the Muntz-Szasz theorem: If 0 < p1 < p2 < p3 < ..., then the span
> of {1, x^p1, x^p2, ...} is dense in C[0,1] iff 1/p1 + 1/p2 + ... = oo.
> (There is a nice proof of this in Rudin's Real and Complex Analysis.)
Thank you both!
Michael Hamm
AM, Math, Wash. U. St. Louis
msh210@math.wustl.edu Standard disclaimers:
<a href="http://math.wustl.edu/~msh210/">http://math.wustl.edu/~msh210/</a> ... legal.html
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