Bernoulli's inequality

Bernoulli's inequality states that (1+x)^r ≥ 1+rx for x ≥ −1 and real r ≥ 1, with the familiar integer-exponent form for nonnegative integers.

Mathematical induction: A proof method that establishes a base case and then proves each case follows from its predecessor. Induction proves the inequality successively for nonnegative integer exponents.
Compound interest: Interest calculated repeatedly on principal together with previously accumulated interest. Bernoulli's inequality bounds the growth of (1+r/n)^n in discrete compounding estimates.
Real number: An element of the number system representing points on a continuous number line. The generalized statement allows real exponents, not only integer ones.
Weighted arithmetic–geometric mean inequality: An inequality stating that a weighted arithmetic mean of positive numbers is at least their weighted geometric mean. Both inequalities compare powers with linear expressions, though their hypotheses and forms differ.
Convex function: A function whose graph lies below each chord joining two points on its graph. For exponents at least one, convexity places the power curve above its tangent at one.
Exponential function: The function exp(x), equal to its own derivative and inverse to the natural logarithm. Taking powers with exponent n and increment x/n connects the inequality to exponential growth bounds.
Integer: A whole number, positive, negative, or zero. The classical elementary version restricts the exponent to a nonnegative integer.
Taylor's theorem: A theorem approximating a sufficiently smooth function by a polynomial with a controlled remainder. Bernoulli's bound is a one-sided first-order estimate for a power function.
Tangent line: A line that meets a smooth curve at a point with the same local slope. The bound 1+rx is the tangent line to (1+x)^r at x=0.
Limit comparison test: A test for comparing convergence of two series with term ratios approaching a positive finite limit. Its estimates can use Bernoulli's inequality to control terms involving powers near one.

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