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Re: Re: which values of m satisfies the inequality

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104274] Re: [mg104231] Re: which values of m satisfies the inequality
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 25 Oct 2009 01:10:31 -0400 (EDT)
  • References: <hbr59l$rh6$1@smc.vnet.net> <200910240641.CAA07606@smc.vnet.net> <34417DDF-DE3E-48D9-8EB4-A4239A59F9E3@mimuw.edu.pl> <FB115BF502FD4D52818DA3A62B7F8B6B@Dell>

On 24 Oct 2009, at 23:18, David W. Cantrell wrote:

> 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
>

Yes, but this is not "if and only if".

(1 + x)^(m + 1) > (1 + x^m)*2^m /. {x -> 10, m -> 4}

  True

but 4 >3.358...

See my post in this thread.

Andrzej





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