Talk:Bernoulli's inequality
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[edit] asdf
In generalized inequality: if x=0, both inequalities are valid. -- anon
This has been fixed. -- Toby Bartels
[edit] a little bit more
1-x<e^(-x)
Jackzhp 20:00, 6 November 2006 (UTC)
[edit] This entry should be for real values of r, not just for integer values
Bernoulli's inequality true for all real values of r ≥ 1. This is important for many applications, so it is less useful to just describe it for integer values of r. And I do not think that it is worthwhile including an induction proof, which only gives the result for integer values of r.
Here's a quick proof for all real values of r ≥ 1. Unfortunately, I don't have the Wikipedia skills to directly edit this entry, so maybe someone else can do it.
Let f(x) = (1+x)^r - 1 - rx. The extended mean value theorem says that
f(x) = f(0) + f'(0)x + (1/2)f(y)x^2
for some y between 0 and x. But f(0) = 0 and f'(0) = 0, so
f(x) = r(r-1)(1+y)^{r-2}x^2.
for some y between 0 and x. It is clear that this last expression is nonnegative if x > -1 and r ≥ 1. —Preceding unsigned comment added by 76.118.41.44 (talk) 14:59, 4 November 2007 (UTC)
[edit] For n integer values the inequality also applies for x>=-2
For some reason this is not mentioned anywhere, but upon trying to prove the inequality I stumbled upon that fact. It can be easily shown that for even n values the inequality applies for all real x values, and for odd n's, for all x>=-2.
- Maybe it is not mentioned because for non-integer n values - a^n is meaningless for negative a values (-2<=x<-1).
[Shir Peled] —Preceding unsigned comment added by 128.139.226.37 (talk) 18:09, 16 December 2007 (UTC)
