Binomial Theorem/Approximations

Theorem

Consider the General Binomial Theorem:

$\paren {1 + x}^\alpha = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots$

When $x$ is small it is often possible to neglect terms in $x$ higher than a certain power of $x$, and use what is left as an approximation to $\paren {1 + x}^\alpha$.

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First Order

When $x$ is sufficiently small that $x^2$ can be neglected then:

$\paren {1 + x}^\alpha \approx 1 + \alpha x$

and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} } 2 x^2$

Second Order

When $x$ is sufficiently small that $x^3$ can be neglected then:

$\paren {1 + x}^\alpha \approx 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } 2 x^2$

and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 3} } 6 x^3$

Examples

Arbitrary Example $1$

$\paren {1 \cdotp 0 6}^{1/3} \approx 1 \cdotp 019613$

to $6$ decimal places.

Arbitrary Example $2$

$\sqrt {25 \cdotp 1} \approx 5 \cdotp 0100$

to $4$ decimal places.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Approximations