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Re: Proving inequalities with Mathematica

On 29 Nov 2004, at 15:22, Toshiyuki (Toshi) Meshii wrote:

> Hi,
> I was wondering whether Mathematica is useful for proving a problem of
> inequality.
> My problem is as follows:
> Let An and Bn (n=1,2,3...) be real sequences.
> Some characteristics of these sequences are known.
> i) Abs[An+1/An] < 1
> ii) Abs[Bn+1/Bn] < 1
> iii) Sum[An, {1,Infinity}] = 0
> iv) Sum[An*Bn, {1, Infinity}] = alpha (note: a real number)
> Then I want to prove with Mathematica that
> 0 < Abs[alpha] < Abs[A1*B1]
> Does anyone have an idea?
> -Toshi

First of all, you have written your condition as

> Abs[An+1/An] < 1

Presumably you meant n+1 to be a subscript (if not the condition is 
always false).
Assuming that,  Mathematica can certainly be helpful in proving your 
claim to be false. Here is an example.

We define the sequence A[n] as follows:

A[1] = -1; A[2] = Log[2];
A[n_] = (-1)^(n - 1)/(n - 1);

You can check that the condition Abs[A[n+1]/A[n]]<1 is always 
satisfied. We also define the sequence B[n] by
B[i_] := A[i]
so that the condition


is satisifed.

We check that (iii) is satisfied:



Let's now compute alpha:



Let's see if your inequality holds:



Oops ...

Andrzej Kozlowski
Chiba, Japan

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