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Re: Re: which values of m satisfies the inequality
*To*: mathgroup at smc.vnet.net
*Subject*: [mg104257] Re: [mg104231] Re: which values of m satisfies the inequality
*From*: "David W. Cantrell" <DWCantrell at sigmaxi.net>
*Date*: Sun, 25 Oct 2009 01:07:20 -0400 (EDT)
*References*: <hbr59l$rh6$1@smc.vnet.net> <200910240641.CAA07606@smc.vnet.net> <34417DDF-DE3E-48D9-8EB4-A4239A59F9E3@mimuw.edu.pl>
On 24 Oct 2009, at 07:24, Andrzej Kozlowski wrote:
> On 24 Oct 2009, at 15:41, David W. Cantrell wrote:
>> barefoot gigantor <barefoot1980 at gmail.com> wrote:
>>> for what value or interval of m (-infinity < m < infinity) the
>>> following is valid
>>>
>>> (1+x)^(m+1) > (1+x^m)* 2^m
>>>
>>> here x >=1
>>
>> I shall assume that you really meant to have strictly x > 1.
>> [That's because, if x = 1, then (1+x)^(m+1) = (1+x^m) * 2^m,
>> regardless of m.]
>>
>> Answer:
>>
>> For x > 1 and m >= -1,
>>
>> (1+x)^(m+1) > (1+x^m) * 2^m
>
> Hmm...
>
> (1 + x)^(m + 1) > (1 + x^m)*2^m /. {x -> 10, m -> 6}
>
> False
>
Thanks, Andrzej! Yes, I mistakenly left off the upper bound for m.
I should have written:
For x > 1 and -1 <= m < c where c = 3.358...,
(1+x)^(m+1) > (1+x^m) * 2^m
[barefoot: If it's important to you to have constant c determined with
greater accuracy, just ask.]
David
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