
Re: subseries of harmonic series converges iff ...
Posted:
Nov 21, 2004 9:11 AM


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 MuntzSzasz 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

