Re: How tp prove this proposition?

From: Zdislav V. Kovarik (kovarik_at_mcmaster.ca)
Date: 10/14/04 

Date: Thu, 14 Oct 2004 16:26:53 -0400

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).

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