- Bernoulli's inequality
real analysis, Bernoulli's inequality is an inequalitythat approximates exponentiations of 1 + "x".
The inequality states that:for every
integer"r" ≥ 0 and every real number"x" > −1. If the exponent "r" is even, then the inequality is valid for "all" real numbers "x". The strict version of the inequality reads:for every integer "r" ≥ 2 and every real number "x" ≥ −1 with "x" ≠ 0.
Bernoulli's inequality is often used as the crucial step in the proof of other inequalities. It can itself be proved using
mathematical induction, as shown below.
Proof of the inequality
For :is equivalent to which is true as required.
Now suppose the statement is true for ::Then it follows that: (by hypothesis, since )
However, as (since ), it follows that , which means the statement is true for as required.
By induction we conclude the statement is true for all
The exponent "r" can be generalized to an arbitrary real number as follows: if "x" > −1, then:for "r" ≤ 0 or "r" ≥ 1, and :for 0 ≤ "r" ≤ 1.This generalization can be proved by comparing
derivatives.Again, the strict versions of these inequalities require "x" ≠ 0 and "r" ≠ 0, 1.
The following inequality estimates the "r"-th power of 1 + "x" from the other side. For any real numbers "x", "r" > 0, one has:where "e" = 2.718....This may be proved using the inequality (1 + 1/"k")"k" < "e".
* [http://demonstrations.wolfram.com/BernoulliInequality/ Bernoulli Inequality] by Chris Boucher,
The Wolfram Demonstrations Project.
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