Bernoulli Inequality Calculator
The Bernoulli Inequality calculator helps students and mathematicians verify the inequality $$(1 + x)^r \geq 1 + rx$$, which holds for real numbers $$x \geq -1$$ and integers $$r \geq 0$$. This tool is particularly useful for learning and validating mathematical proofs and concepts involving inequalities and exponential growth.
Bernoulli Inequality Calculator
Bernoulli Inequality Formula
This inequality states that for any real number $$x \geq -1$$ and any integer $$r \geq 0$$, the expression $$(1 + x)^r$$ is always greater than or equal to $$1 + rx$$. It’s commonly used in mathematical analysis and proofs, particularly when estimating powers or dealing with compound interest and exponential growth.
The Bernoulli Inequality is a fundamental result in algebra and analysis, often applied in contexts such as approximation theory, algorithm complexity, and financial modeling. For example, it provides a lower bound for expressions involving exponentiation. If $$x = 0.1$$ and $$r = 3$$, the left-hand side becomes $$(1 + 0.1)^3 = 1.331$$, while the right-hand side is $$1 + 3 \cdot 0.1 = 1.3$$, confirming the inequality. The calculator simplifies this comparison, allowing you to input values of x and r to instantly check whether the inequality holds.