# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Bernoulli's inequality

(Redirected from Bernoulli inequality)

In mathematics, Bernoulli's inequality states that

$(1+x)^n\geq 1+nx\,$

for every integer n ≥ 0 and every real number x ≥ −1. If n ≥ 0 is even, then the inequality is valid for all real numbers x. The strict version of the inequality reads

$(1+x)^n>1+nx\,$

for every integer n ≥ 2 and every real number x ≥ −1 with x ≠ 0.

The inequality is often used as the crucial step in the proof of other inequalities. It can be proven using mathematical induction.

The following generalizations for real exponents can be proved by comparing derivatives: if x > −1, then

$(1+x)^r\geq 1+rx\,$

for r ≤ 0 or r ≥ 1 and

$(1+x)^r\leq 1+rx\,$

for 0 ≤ r ≤ 1.

## Related inequalities

The following inequality estimates the r-th power of 1+x from the other side. For any x,r > 0 one has

$(1+x)^r < e^{rx}.\,$

03-10-2013 05:06:04