Re: Infrequent Mathematica User

• To: mathgroup at smc.vnet.net
• Subject: [mg47090] Re: Infrequent Mathematica User
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 25 Mar 2004 05:48:21 -0500 (EST)
• References: <163.2d33e8ae.2d90396b@aol.com> <001f01c41166\$1d4ae340\$58868218@we1.client2.attbi.com>
• Sender: owner-wri-mathgroup at wolfram.com

```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

```

• Prev by Date: NDSolve Repeated convergence test failure
• Next by Date: Re: Manipulating the Front End
• Previous by thread: Re: Infrequent Mathematica User
• Next by thread: Re: Infrequent Mathematica User