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MathGroup Archive 2004

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Re: Re: Infrequent Mathematica User

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47299] Re: [mg47244] Re: Infrequent Mathematica User
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 2 Apr 2004 03:31:50 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

On 2/4/04, Andrzej Kozlowski wrote:

>Actually one can use Paul's argument to prove 
>the following stronger statement:
>
>Sum[(Subscript[x,i]/(1 + Sum[Subscript[x,j]^2, {j, i}]))^2, {i, n}] < 1
>
>for every positive integer n.
>
>It is easy to see that this implies the 
>inequality in the original problem (use 
>Schwarz's inequality).

MathWorld only gives the integral form at

   http://mathworld.wolfram.com/SchwarzsInequality.html

The required form of the inequality is at

   http://mathworld.wolfram.com/CauchysInequality.html

>Moreover, the proof is easier since the inductive step is now trivial.

Nice.

>In addition, the inequality leads to some 
>intriguing observations and also to what looks 
>like a bug in Limit (?)

Actually, I think the problem is with Series. I've submitted a bug report.

>The inequality implies that the sums, considered 
>as functions on the real line, are bounded and 
>attain their maxima. So it is natural to 
>consider the functions f[n] (obtained by setting 
>all the Subscript[x,i] = Subscript[x,j)]
>
>f[n_][x_] := NSum[(x/(i*x^2 + 1))^2, {i, 1, n}]
>
>It is interesting to look at:
>
>plots = Table[
>     Plot[f[n][x], {x, -1, 1}, DisplayFunction -> Identity], {n, 1, 10}];
>
>Show[plots, DisplayFunction -> $DisplayFunction]
>
>The f[n] of course also bounded by 1 and so in the limit we have the function:
>
>f[x_] = Sum[(x/(i*x^2 + 1))^2, {i, 1, Infinity}]
>
>PolyGamma[1, (x^2 + 1)/x^2]/x^2
>
>which also ought to be bounded bu 1.
>
>Plotting the graph of this, e.g.
>
>Plot[f[x], {x, -0.1, 0.1}]
>
>shows a maximum value 1 at 0 (where the function 
>is not defined), however Mathematica seems to 
>give the wrong limit:
>
>Limit[f[x],x->0]
>
>-?

Series also gives an incorrect result (I think Limit is using this).

Cheers,
Paul


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