Bernoulli's inequality

In real analysis, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + "x".

The inequality states that:$\left(1 + x\right)^r geq 1 + rx!$for every integer "r" &ge; 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:$\left(1 + x\right)^r > 1 + rx!$for every integer "r" &ge; 2 and every real number "x" &ge; −1 with "x" &ne; 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 $r=0,,$:$\left(1+x\right)^0 ge 1+0x$is equivalent to $1ge 1$ which is true as required.

Now suppose the statement is true for $r=k$::$\left(1+x\right)^k ge 1+kx.$Then it follows that:$\left(1+x\right)\left(1+x\right)^k ge \left(1+x\right)\left(1+kx\right)$ (by hypothesis, since $\left(1+x\right)ge 0$)

:

However, as $1+\left(k+1\right)x + kx^2 ge 1+\left(k+1\right)x$ (since $kx^2 ge 0$), it follows that $\left(1+x\right)^\left\{k+1\right\} ge 1+\left(k+1\right)x$, which means the statement is true for $r=k+1$ as required.

By induction we conclude the statement is true for all $rge 0.$

Generalization

The exponent "r" can be generalized to an arbitrary real number as follows: if "x" > −1, then:$\left(1 + x\right)^r geq 1 + rx!$for "r" &le; 0 or "r" &ge; 1, and :$\left(1 + x\right)^r leq 1 + rx!$for 0 &le; "r" &le; 1.This generalization can be proved by comparing derivatives.Again, the strict versions of these inequalities require "x" &ne; 0 and "r" &ne; 0, 1.

Related inequalities

The following inequality estimates the "r"-th power of 1 + "x" from the other side. For any real numbers "x", "r" &gt; 0, one has:$\left(1 + x\right)^r le e^\left\{rx\right\},!$where "e" = 2.718....This may be proved using the inequality (1 + 1/"k")"k" < "e".

References

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* [http://demonstrations.wolfram.com/BernoulliInequality/ Bernoulli Inequality] by Chris Boucher, The Wolfram Demonstrations Project.

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