binomial approximation

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

(1 + x)^{α} \approx 1 + α x .

It is valid when

| x | < 1

and

| α x | ≪ 1

where

x

and

α

may be real or complex numbers.

The benefit of this approximation is that

α

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.

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

x > - 1

and

α \geq 1

Derivations

Using linear approximation

The function

f (x) = (1 + x)^{α}

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

f^{'} (x) = α (1 + x)^{α - 1}

and so

f^{'} (0) = α .

Thus

f (x) \approx f (0) + f^{'} (0) (x - 0) = 1 + α x .

By Taylor's theorem, the error in this approximation is equal to

\frac{α (α - 1) x^{2}}{2} \cdot (1 + ζ)^{α - 2}

for some value of

ζ

that lies between 0 and x. For example, if

x < 0

and

α \geq 2

, the error is at most

\frac{α (α - 1) x^{2}}{2}

. In little o notation, one can say that the error is

o (| x |)

, meaning that

\lim_{x \to 0} \frac{e r r o r}{| x |} = 0

Using Taylor series

The function

f (x) = (1 + x)^{α}

where

x

and

α

may be real or complex can be expressed as a Taylor series about the point zero.

\begin{array}{l} 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} + \dots \\ (1 + x)^{α} & = 1 + α x + \frac{1}{2} α (α - 1) x^{2} + \frac{1}{6} α (α - 1) (α - 2) x^{3} + \frac{1}{24} α (α - 1) (α - 2) (α - 3) x^{4} + \dots \end{array}

| x | < 1

and

| α x | ≪ 1

, then the terms in the series become progressively smaller and it can be truncated to

(1 + x)^{α} \approx 1 + α 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

| α 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

| x | ≪ 1

is a sufficient condition for the binomial approximation. A simple counterexample is to let

x = 10^{- 6}

and

α = 10^{7}

. In this case

(1 + x)^{α} > 22, 000

but the binomial approximation yields

1 + α x = 11

. For small

| x |

but large

| α x |

, a better approximation is:

(1 + x)^{α} \approx e^{α x} .

Example

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

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

where

a

and

b

are real but

a ≫ b

.

The mathematical form for the binomial approximation can be recovered by factoring out the large term

a

and recalling that a square root is the same as a power of one half.

\begin{array}{l} \frac{1}{\sqrt{a + b}} - \frac{1}{\sqrt{a - b}} & = \frac{1}{\sqrt{a}} ({(1 + \frac{b}{a})}^{- 1 / 2} - {(1 - \frac{b}{a})}^{- 1 / 2}) \\ \approx \frac{1}{\sqrt{a}} ((1 + (- \frac{1}{2}) \frac{b}{a}) - (1 - (- \frac{1}{2}) \frac{b}{a})) \\ \approx \frac{1}{\sqrt{a}} (1 - \frac{b}{2 a} - 1 - \frac{b}{2 a}) \\ \approx - \frac{b}{a \sqrt{a}} \end{array}

Evidently the expression is linear in

b

when

a ≫ b

which is otherwise not obvious from the original expression.

Generalization

Binomial series
While the binomial approximation is linear, it can be generalized to a quadratic approximation keeping the second term in the Taylor series:

(1 + x)^{α} \approx 1 + α x + (α / 2) (α - 1) x^{2}

Applied to the square root, it results in:

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

Quadratic example

Consider the expression:

(1 + ϵ)^{n} - (1 - ϵ)^{- n}

where

| ϵ | < 1

and

| n ϵ | ≪ 1

. If only the linear term from the binomial approximation is kept

(1 + x)^{α} \approx 1 + α x

then the expression unhelpfully simplifies to zero

\begin{array}{l} (1 + ϵ)^{n} - (1 - ϵ)^{- n} & \approx (1 + n ϵ) - (1 - (- n) ϵ) \\ \approx (1 + n ϵ) - (1 + n ϵ) \\ \approx 0 . \end{array}

While the expression is small, it is not exactly zero.

So now, keeping the quadratic term:

\begin{array}{l} (1 + ϵ)^{n} - (1 - ϵ)^{- n} & \approx (1 + n ϵ + \frac{1}{2} n (n - 1) ϵ^{2}) - (1 + (- n) (- ϵ) + \frac{1}{2} (- n) (- n - 1) (- ϵ)^{2}) \\ \approx (1 + n ϵ + \frac{1}{2} n (n - 1) ϵ^{2}) - (1 + n ϵ + \frac{1}{2} n (n + 1) ϵ^{2}) \\ \approx \frac{1}{2} n (n - 1) ϵ^{2} - \frac{1}{2} n (n + 1) ϵ^{2} \\ \approx \frac{1}{2} n ϵ^{2} ((n - 1) - (n + 1)) \\ \approx - n ϵ^{2} \end{array}

This result is quadratic in

ϵ

which is why it did not appear when only the linear terms in

ϵ

were kept.

References

Griffiths, D., 1999, Introduction to Electrodynamics, Pearson Education, Inc., Third, 146–148

Category:Factorial and binomial topics
Category:Approximations

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