Binomial Theorem/Approximations/2nd Order

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 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:

$\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } 6 x^3$


Proof


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Sources

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