Bernoulli's inequality
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In mathematics, Bernoulli's inequality is an inequality that 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:
Proof: For r=0, 
equals 
which is true as required.
Now suppose the statement is true for r=k: 
Then it follows that 
(by hypothesis, since 
) 
(since 
) So it follows that 
, which means the statement is true for r=k+1 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.
[edit] Related inequalities
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.
