Re: Re: Infrequent Mathematica User
- To: mathgroup at smc.vnet.net
- Subject: [mg47293] Re: [mg47244] Re: Infrequent Mathematica User
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Fri, 2 Apr 2004 03:31:17 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
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). Moreover, the proof is easier since
the inductive step is now trivial.In addition, the inequality leads to
some intriguing observations and also to what looks like a bug in Limit
(?)
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]
-°
?
Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/
On 31 Mar 2004, at 16:59, Bobby R. Treat wrote:
> OK, as I suspected, there's apparently no standard inequality such as
> the Cauchy Sum Inequality that applies (at least, not directly). There
> couldn't be, because they're all "tight" for every n and all can be
> made strict equalities for some choice of the variables, while neither
> is true for the problem inequality, which is never an equality and is
> tight only in the limit as n->Infinity.
>
> Meanwhile, the "proof" at the link is a proof only if we blindly trust
> Reduce. (I doubt that we always can.) I really liked the
> "straightforward" comment before the last use of Reduce. I had no
> trouble replacing the first use with an actual proof, but the second
> is trickier.
>
> Bobby
>
> Paul Abbott <paul at physics.uwa.edu.au> wrote in message
> news:<c4bdvq$6vn$1 at smc.vnet.net>...
>> In article <c4390g$en3$1 at smc.vnet.net>,
>> drbob at bigfoot.com (Bobby R. Treat) wrote:
>>
>>> If you actually see a way to apply one of the standard inequalities
>>> to
>>> this sum, please share. Nothing obvious springs to mind.
>>
>> For a proof see
>>
>>
>> ftp://physics.uwa.edu.au/pub/Mathematica/MathGroup/InequalityProof.nb
>>
>> Cheers,
>> Paul
>>
>>
>>> 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 P?lya, 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
>>>
>
>
>