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

**References**:**Re: which values of m satisfies the inequality***From:*"David W. Cantrell" <DWCantrell@sigmaxi.net>