Bernoulli Inequality Calculator - Mathematical Induction

Enter value and click on calculate. Result will be displayed.

Formula:
(1 + x) r ≥ 1 + rx

x Value :
Power (r) :
Result:
Bernouli Inequality :

What is a Bernoulli Inequality Calculator?
A Bernoulli Inequality Calculator is a tool that helps verify or apply the Bernoulli Inequality, a fundamental inequality in mathematics. The Bernoulli Inequality states:

(1+x)n≥1+nx

where:

  • x>−1
  • n ≥ 0 is an integer

This inequality shows how powers of sums behave and sets the foundation for many results in calculus and algebra.

Why use a Bernoulli Inequality Calculator?

  • Verification: It checks whether the inequality holds for given x and n.
  • Mathematical Proofs: Bernoulli’s inequality is often used in proofs by induction, limits, and approximations.
  • Real-World Applications: It’s applied in finance (compound interest), computer science (algorithm analysis), and physics (estimation).
  • Simplifying expressions: It helps bound exponential functions and power expressions.

How does a Bernoulli Inequality Calculator work?

  • Takes input values for x and n.
  • Computes both sides of the inequality: (1+x)n and 1+nx.
  • Checks if the inequality (1+x)n≥1+nx holds true.
  • Optionally, it can use mathematical induction to prove the inequality for all n≥0:
    1. Base Case: Show it’s true for n=0 or n=1 .
    2. Inductive Step: Assume it’s true for nn n and prove for n+1 .

When is a Bernoulli Inequality Calculator used?

  • In mathematical induction proofs: To establish the inequality for all positive integers.
  • In calculus and algebra: To bound functions and simplify estimates.
  • In economics and finance: To approximate compound interest and growth rates.
  • In computer science: For analyzing the performance of exponential-time algorithms.