The binomial approximation is useful for approximately calculating powers of sums of a small number and 1. It states that if
Contents
- Derivation using linear approximation
- Derivation using Taylor Series
- Example simplification
- Example keeping the quadratic term
- References
If either
The benefit of this approximation is that
This approximation can be proven several ways including 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
Derivation using linear approximation
The function
is a smooth function for x near 0. Thus, standard linear approximation tools from calculus apply: one has
and so
Thus
Derivation using Taylor Series
The function
where
If
This result from the binomial approximation can always be improved by keeping additional terms from the Taylor Series above. This is especially important when
Sometimes it is wrongly claimed that
Example simplification
Consider the following expression where
The mathematical form for the binomial approximation can be recovered by factoring out the large term
Evidently the expression is linear in
Example keeping the quadratic term
Consider the expression:
where
While the expression is small, it is not exactly zero. It is possible to extract a nonzero approximate solution by keeping the quadratic term in the Taylor Series, i.e.
This result is quadratic in