Using linear approximation The function : f(x) = (1 + x)^{\alpha} is a
smooth function for
x near 0. Thus, standard
linear approximation tools from
calculus apply: one has : f'(x) = \alpha (1 + x)^{\alpha - 1} and so : f'(0) = \alpha. Thus : f(x) \approx f(0) + f'(0)(x - 0) = 1 + \alpha x. By
Taylor's theorem, the error in this approximation is equal to \frac{\alpha(\alpha - 1) x^2}{2} \cdot (1 + \zeta)^{\alpha - 2} for some value of \zeta that lies between 0 and . For example, if x and \alpha \geq 2, the error is at most \frac{\alpha(\alpha - 1) x^2}{2}. In
little o notation, one can say that the error is o(|x|), meaning that \lim_{x \to 0} \frac{\textrm{error}} = 0.
Using Taylor series The function : f(x) = (1+x)^\alpha where x and \alpha may be real or complex can be expressed as a
Taylor series about the point zero. :\begin{align} 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{align} If |x| and |\alpha x| \ll 1, then the terms in the series become progressively smaller and it can be truncated to :(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 |\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 |x| \ll 1 is a sufficient condition for the binomial approximation. A simple counterexample is to let x=10^{-6} and \alpha=10^7. In this case (1+x)^\alpha > 22,000 but the binomial approximation yields 1 + \alpha x = 11. For small |x| but large |\alpha x|, a better approximation is: : (1 + x)^\alpha \approx e^{\alpha x} . == Example ==