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In real analysis, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + ''x''. The inequality states that : for every integer ''r'' ≥ 0 and every real number ''x'' ≥ −1. There is also a generalized version that says for every real number r ≥ 1, and real number x ≥ -1 : while for 0 ≤ ''r'' ≤ 1, and 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, as shown below. ==History== Jacob Bernoulli first published the inequality in his treatise “Positiones Arithmeticae de Seriebus Infinitis” (Basel, 1689), where he used the inequality often.〔(mathematics - First use of Bernoulli's inequality and its name - History of Science and Mathematics Stack Exchange )〕 According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963), p. 177, the inequality is actually due to Sluse in his Mesolabum (1668 edition), Chapter IV "De maximis & minimis".〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bernoulli's inequality」の詳細全文を読む スポンサード リンク
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