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Re: Infrequent Mathematica User
*To*: mathgroup at smc.vnet.net
*Subject*: [mg47149] Re: Infrequent Mathematica User
*From*: drbob at bigfoot.com (Bobby R. Treat)
*Date*: Sat, 27 Mar 2004 01:35:17 -0500 (EST)
*Organization*: http://groups.google.com
*References*: <163.2d33e8ae.2d90396b@aol.com> <001f01c41166$1d4ae340$58868218@we1.client2.attbi.com> <c3ueie$9ti$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
If you actually see a way to apply one of the standard inequalities to
this sum, please share. Nothing obvious springs to mind.
Bobby
Paul Abbott <paul at physics.uwa.edu.au> wrote in message news:<c3ueie$9ti$1 at smc.vnet.net>...
> Jim Dars wrote:
>
> >A Math NG posed the problem:
> >
> > Let x1,x2,...,xn be real numbers. Prove
> > x1/(1+x1^2) + x2/(1+x1^2+x2^2) +...+ xn/(1+x1^2+...+xn^2) < sqrt(n)
>
> To prove this, I would do a search for inequalites, e.g,
>
> http://mathworld.wolfram.com/ChebyshevSumInequality.html
>
> Also, see
>
> Hardy, G. H.; Littlewood, J. E.; and Pólya, G.
> Inequalities, 2nd ed. Cambridge, England:
> Cambridge University Press, pp. 43-44, 1988.
>
> To investigate using Mathematica, I like to use subscripted variables:
>
> s[n_] := Sum[Subscript[x,i]/(1 + Sum[Subscript[x,j]^2, {j, i}]), {i, n}]
>
> If you enter this expression into Mathematica and
> do Cell | Convert to StandardForm (or
> TraditionalForm) you will get a nicely formatted
> expression for the n-th left-hand side of the
> inequality.
>
> Note that you can prove the inequality on a
> case-by-case basis using CylindricalDecomposition
> (in Version 5.0):
>
> CylindricalDecomposition[s[1] > 1, {Subscript[x, 1]}]
>
> CylindricalDecomposition[s[2] > Sqrt[2], {Subscript[x, 1], Subscript[x, 2]}]
>
> and so on. This may not seem convincing, but see
> what happens if the you change the inequality:
>
> CylindricalDecomposition[s[2] > 1/2, {Subscript[x, 1], Subscript[x, 2]}]
>
> >To get a feel for the problem, and maybe spark
> >an idea, I hoped to look at some few early
> >maximum values. However, these proved difficult
> >to come by.
>
> NMaximize is the way to go:
>
> Table[NMaximize[s[n], Table[{Subscript[x, i], -5, 5}, {i, n}]], {n, 6}]
>
> Cheers,
> Paul
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