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In mathematics, Bernoulli's inequality states that
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.
for r ≤ 0 or r ≥ 1 and
for 0 ≤ r ≤ 1.
The following inequality estimates the r-th power of 1+x from the other side. For any x,r > 0 one has
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