Binomial approximation

The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that

${\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}$

It is valid when ${\displaystyle |x|<1}$ and ${\displaystyle |\alpha x|\ll 1}$ where ${\displaystyle x}$ and ${\displaystyle \alpha }$ may be real or complex numbers.

The benefit of this approximation is that ${\displaystyle \alpha }$ is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics.^[1]

The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever ${\displaystyle x>-1}$ and ${\displaystyle \alpha \geq 1}$.

Derivations

Using linear approximation

The function

${\displaystyle f(x)=(1+x)^{\alpha }}$

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

${\displaystyle f'(x)=\alpha (1+x)^{\alpha -1}}$

and so

${\displaystyle f'(0)=\alpha .}$

Thus

${\displaystyle f(x)\approx f(0)+f'(0)(x-0)=1+\alpha x.}$

By Taylor's theorem, the error in this approximation is equal to ${\textstyle {\frac {\alpha (\alpha -1)x^{2}}{2}}\cdot (1+\zeta )^{\alpha -2}}$ for some value of ${\displaystyle \zeta }$ that lies between 0 and x. For example, if ${\displaystyle x<0}$ and ${\displaystyle \alpha \geq 2}$, the error is at most ${\textstyle {\frac {\alpha (\alpha -1)x^{2}}{2}}}$. In little o notation, one can say that the error is ${\displaystyle o(|x|)}$, meaning that ${\textstyle \lim _{x\to 0}{\frac {\textrm {error}}{|x|}}=0}$.

Using Taylor series

The function

${\displaystyle f(x)=(1+x)^{\alpha }}$

where ${\displaystyle x}$ and ${\displaystyle \alpha }$ may be real or complex can be expressed as a Taylor series about the point zero.

${\displaystyle {\begin{aligned}f(x)&=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}\\f(x)&=f(0)+f'(0)x+{\frac {1}{2}}f''(0)x^{2}+{\frac {1}{6}}f'''(0)x^{3}+{\frac {1}{24}}f^{(4)}(0)x^{4}+\cdots \\(1+x)^{\alpha }&=1+\alpha x+{\frac {1}{2}}\alpha (\alpha -1)x^{2}+{\frac {1}{6}}\alpha (\alpha -1)(\alpha -2)x^{3}+{\frac {1}{24}}\alpha (\alpha -1)(\alpha -2)(\alpha -3)x^{4}+\cdots \end{aligned}}}$

If ${\displaystyle |x|<1}$ and ${\displaystyle |\alpha x|\ll 1}$, then the terms in the series become progressively smaller and it can be truncated to

${\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}$

This result from the binomial approximation can always be improved by keeping additional terms from the Taylor series above. This is especially important when ${\displaystyle |\alpha x|}$ starts to approach one, or when evaluating a more complex expression where the first two terms in the Taylor series cancel (see example).

Sometimes it is wrongly claimed that ${\displaystyle |x|\ll 1}$ is a sufficient condition for the binomial approximation. A simple counterexample is to let ${\displaystyle x=10^{-6}}$ and ${\displaystyle \alpha =10^{7}}$. In this case ${\displaystyle (1+x)^{\alpha }>22,000}$ but the binomial approximation yields ${\displaystyle 1+\alpha x=11}$. For small ${\displaystyle |x|}$ but large ${\displaystyle |\alpha x|}$, a better approximation is:

${\displaystyle (1+x)^{\alpha }\approx e^{\alpha x}.}$

Example

The binomial approximation for the square root, ${\displaystyle {\sqrt {1+x}}\approx 1+x/2}$, can be applied for the following expression,

${\displaystyle {\frac {1}{\sqrt {a+b}}}-{\frac {1}{\sqrt {a-b}}}}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are real but ${\displaystyle a\gg b}$.

The mathematical form for the binomial approximation can be recovered by factoring out the large term ${\displaystyle a}$ and recalling that a square root is the same as a power of one half.

${\displaystyle {\begin{aligned}{\frac {1}{\sqrt {a+b}}}-{\frac {1}{\sqrt {a-b}}}&={\frac {1}{\sqrt {a}}}\left(\left(1+{\frac {b}{a}}\right)^{-1/2}-\left(1-{\frac {b}{a}}\right)^{-1/2}\right)\\&\approx {\frac {1}{\sqrt {a}}}\left(\left(1+\left(-{\frac {1}{2}}\right){\frac {b}{a}}\right)-\left(1-\left(-{\frac {1}{2}}\right){\frac {b}{a}}\right)\right)\\&\approx {\frac {1}{\sqrt {a}}}\left(1-{\frac {b}{2a}}-1-{\frac {b}{2a}}\right)\\&\approx -{\frac {b}{a{\sqrt {a}}}}\end{aligned}}}$

Evidently the expression is linear in ${\displaystyle b}$ when ${\displaystyle a\gg b}$ which is otherwise not obvious from the original expression.

Generalization

While the binomial approximation is linear, it can be generalized to keep the quadratic term in the Taylor series:

${\displaystyle (1+x)^{\alpha }\approx 1+\alpha x+(\alpha /2)(\alpha -1)x^{2}}$

Applied to the square root, it results in:

${\displaystyle {\sqrt {1+x}}\approx 1+x/2-x^{2}/8.}$

Quadratic example

Consider the expression:

${\displaystyle (1+\epsilon )^{n}-(1-\epsilon )^{-n}}$

where ${\displaystyle |\epsilon |<1}$ and ${\displaystyle |n\epsilon |\ll 1}$. If only the linear term from the binomial approximation is kept ${\displaystyle (1+x)^{\alpha }\approx 1+\alpha x}$ then the expression unhelpfully simplifies to zero

${\displaystyle {\begin{aligned}(1+\epsilon )^{n}-(1-\epsilon )^{-n}&\approx (1+n\epsilon )-(1-(-n)\epsilon )\\&\approx (1+n\epsilon )-(1+n\epsilon )\\&\approx 0.\end{aligned}}}$

While the expression is small, it is not exactly zero. So now, keeping the quadratic term:

${\displaystyle {\begin{aligned}(1+\epsilon )^{n}-(1-\epsilon )^{-n}&\approx \left(1+n\epsilon +{\frac {1}{2}}n(n-1)\epsilon ^{2}\right)-\left(1+(-n)(-\epsilon )+{\frac {1}{2}}(-n)(-n-1)(-\epsilon )^{2}\right)\\&\approx \left(1+n\epsilon +{\frac {1}{2}}n(n-1)\epsilon ^{2}\right)-\left(1+n\epsilon +{\frac {1}{2}}n(n+1)\epsilon ^{2}\right)\\&\approx {\frac {1}{2}}n(n-1)\epsilon ^{2}-{\frac {1}{2}}n(n+1)\epsilon ^{2}\\&\approx {\frac {1}{2}}n\epsilon ^{2}((n-1)-(n+1))\\&\approx -n\epsilon ^{2}\end{aligned}}}$

This result is quadratic in ${\displaystyle \epsilon }$ which is why it did not appear when only the linear terms in ${\displaystyle \epsilon }$ were kept.

References