Re: How tp prove this proposition?
From: Julian V. Noble (jvn_at_virginia.edu)
Date: 10/17/04
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Date: Sun, 17 Oct 2004 15:34:33 -0400
"Zdislav V. Kovarik" wrote:
>
> On Thu, 14 Oct 2004, Julian V. Noble wrote:
>
> > ChunAn wrote:
> > >
> > > How to prove: exp(rT) > (1+r/m)^(mT)
> > >
> > > Many thanks,
> > >
> > > ChunAn
> >
> > I assume r/m > -1 or the problem makes no sense (for mT not an integer).
> >
> > Take the (mT)'th root of both sides and note that with r/m = x, you now
> > have to prove
> >
> > exp(x) > 1 + x, for x > -1 .
> >
> > Let me suggest the Mean Value Theorem. It is easier if you
> > treat the cases
> >
> > I: x > 0
> >
> > II: -1 < x < 0
> >
> > separately.
> >
> >
> > --
> > Julian V. Noble
> > Professor Emeritus of Physics
> > jvn@lessspamformother.virginia.edu
> > ^^^^^^^^^^^^^^^^^^
> > http://galileo.phys.virginia.edu/~jvn/
> >
> > "For there was never yet philosopher that could endure the toothache
> > patiently." -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
> >
> There is a one-shot proof, based on the identity
>
> exp(-x) * (1+x) = 1 - x^2 * integral [0 to 1] t * exp(-x*t) dt
>
> so that 1 + x < exp(x) for all real x.
>
> Cheers, ZVK(Slavek).
Very nice.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"For there was never yet philosopher that could endure the
toothache patiently."
-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
- Previous message: Jan C. Hoffmann: "Re: solve a catenary (cosh) system"
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