Binomial approximation

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The binomial approximation is useful for approximately calculating powers of sums of a small number and 1. It states that if is a real number close to 0 and is a real number, then

This approximation can be obtained by using the binomial theorem and ignoring the terms beyond the first two.

By Bernoulli's inequality, the left-hand side of this relation is greater than or equal to the right-hand side whenever and .

Derivation using linear approximation[edit]

The function

is a smooth function for x near 0. Thus, standard linear approximation tools from calculus apply: one has

and so

Thus