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In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants: [1] Integer exponent. Case 1: for every integer and real number . The inequality is strict if and . Case 2: for every integer and every real number . [2]
Ilustración de la desigualdad de Bernoulli para n =3. Aquí, la gráfica roja corresponde a (1+ x) 3 y ésta nunca es menor que la gráfica azul, correspondiente a 1+3 x. La desigualdad de Bernoulli es aquella que se establece entre números reales. 1 . y la igualdad se obtiene si y sólo si x =0 o n=1.
23 de ago. de 2022 · Theorem. Let x ∈R x ∈ R be a real number such that x> −1 x> − 1. Let n ∈ Z≥0 n ∈ Z ≥ 0 be a positive integer. Then: (1 + x)n ≥ 1 + nx (1 + x) n ≥ 1 + n x.
10 de oct. de 2024 · The Bernoulli inequality states (1+x)^n>1+nx, (1) where x>-1!=0 is a real number and n>1 an integer. This inequality can be proven by taking a Maclaurin series of (1+x)^n, (2) Since the series terminates after a finite number of terms for integral n, the Bernoulli inequality for x>0 is obtained by truncating after the first-order term.
25 de may. de 2024 · Bernoulli’s inequality (named after Jacob Bernoulli) approximates exponentiations of (1 + x) or more simply gives the approximation of (1 + x) n. Mathematically, the inequality is written as: (1 + x) n ≥ 1 + nx, here ‘x’ is any real number greater than -1, and ‘n’ is any integer greater than 1.
My favorite way of proving Bernoulli is to use Jensen inequality. First of all, the inequality is trivial if $1+nx\leq 0$. So suppose that $1+nx>0$.
En matemáticas, la desigualdad de Bernoulli (llamada así por Jacob Bernoulli) es una desigualdad que se aproxima a las exponenciaciones de 1 + x. A menudo se emplea en el análisis real. Tiene varias variantes útiles: +) + para cada entero r≥ 0 y número real x 1.-
