Re: Binomial thm ===> Bernoulli's ineq?


"Nick" <tulse04-news1@xxxxxxxxxxx> wrote in message
news:nIKdncW-jMYsqkbYnZ2dnUVZ8qO3nZ2d@xxxxxxxxx

"José Carlos Santos" <jcsantos@xxxxxxxx> wrote in message
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On 20-02-2007 12:28, kj wrote:

One version of Bernoulli's inequality is

(1+x)^n >= 1 + n*x, for all x>=-1 and all positive integers n.

I found a webpage saying that it can be proved either by induction
or "directly from the binomial theorem", but it gave no details on
the latter. I get the induction proof, and I see clearly how the
inequality follows easily from the binomial theorem in the case
that x>=0 (because in this case, the sum of the binomial expansion
terms above the linear one is clearly non-negative).

But I don't see how Bernoulli's inequality follows from the binomial
theorem when -1<=x< 0.

Does that web page state the inequality for x > -1 or just for x >= 0?

It states it for x >= -1, in fact. It's here:

http://pirate.shu.edu/projects/reals/infinity/proofs/bernoulli.html

Actually, it is here:

http://pirate.shu.edu/projects/reals/infinity/proofs/bernoull.html

Looking at this I had to do a lot of navigation to find out who was the
author. Clearly it is not the university - universities don't publish such
things themselves.

Finally I discovered it at http://pirate.shu.edu/projects/reals/index.html

"Interactive Real Analysis
An Interactive Textbook
(c) 1994 - 1996 Bert G. Wachsmuth
Release 0.9, August 16, 1996"

I don't see where the webpage
http://pirate.shu.edu/projects/reals/infinity/proofs/bernoull.html says
that it can be proved "directly from the binomial theorem".

Doing a search I discover that this statement is contained in
http://pirate.shu.edu/~wachsmut/ira/infinity/proofs/bernoull.html by the
same author.

In fact, I see that it was discussed in sci.math.research in March 2006. See

http://groups.google.to/group/sci.math.research/tree/browse_frm/month/2006-03/486f9ff0d105cd46?rnum=161&_done=%2Fgroup%2Fsci.math.research%2Fbrowse_frm%2Fmonth%2F2006-03%3F

where Jose Carlos Santos was one of the posters.

Nick


.

Relevant Pages

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    (sci.math)
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    ... I found a webpage saying that it can be proved either by induction ... or "directly from the binomial theorem", but it gave no details on ... But I don't see how Bernoulli's inequality follows from the binomial ...
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