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Topic: subseries of harmonic series converges iff ...
Replies: 5   Last Post: Nov 21, 2004 9:11 AM

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 Michael Hamm Posts: 84 Registered: 12/6/04
Re: subseries of harmonic series converges iff ...
Posted: Nov 21, 2004 9:11 AM

in part:
&gt; &gt; &gt;I recall reading a theorem once that says
&gt; &gt; &gt; For a subset S of Z+ , the following are equivalent:
&gt; &gt; &gt; * \Sum_{s \in S} 1/s &lt; \infty
&gt; &gt; &gt; * Something else.
&gt; &gt; &gt;
&gt; &gt; &gt;Does anyone know what that something else is?
&gt; &gt;
&gt; &gt; I was hoping someone else would give the correct answer but I was
&gt; &gt; under the impression there's a correct "Something else" of the form,
&gt; &gt; * { x^s, s \in S } is a basis for C[0,1]
&gt; &gt; (the set of continuous functions on the interval). Or something like that.

&gt; This is the Muntz-Szasz theorem: If 0 &lt; p1 &lt; p2 &lt; p3 &lt; ..., then the span
&gt; of {1, x^p1, x^p2, ...} is dense in C[0,1] iff 1/p1 + 1/p2 + ... = oo.
&gt; (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

Date Subject Author
11/18/04 Michael Hamm
11/19/04 Robert Israel
11/19/04 Dave Rusin