Bernoulli's inequality

In mathematics we mean by the Bernoulli's inequality, a simple but important inequality with which a power function can be estimated down.

For any real number, and any non-negative integer

Named the inequality after the Swiss mathematician Jacob Bernoulli.

Evidence

Bernoulli's inequality can be proved by mathematical induction. The induction base is met:

As induction hypothesis now applies to, and. Then it follows by the induction hypothesis and

After the induction principle the assertion holds for all.

Assertion:

Real for all.

Proof: First, it is defined by

Then by Bernoulli's inequality

However, it is

Then so is

And ultimately

Also as Bernoulli's inequality the following inequality is called the " strictly greater ", " greater or equal " is used instead of one:

For all real numbers, and all natural numbers

The proof can also be by induction on the same pattern as the proof for the phrase " greater than" perform with.

For real exponents, the following generalizations can be demonstrated by comparison of the derivatives: For all

If and

When.

Considering not a power, but a product of different factors, so can be the following generalization by induction show:

If for all, or if for all and.

Substituting this and considered the special case, ie, we obtain the so-called Weierstrass product inequality

Bernoulli's inequality is useful in many assessments. Fix a solid. Then for all. Therefore applies to the Bernoulli's inequality

Because of

Is thus the inequality

Proved.

Using an estimate of the Bernoulli's inequality can the inequality of the arithmetic and geometric means prove inductively.

Exponentiation

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