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Section 7.7 Common Power Series

The following power series for common functions are used so often in approximations in physics, that you should make the extra effort to memorize the first few terms of each one.

\begin{align*} \sin(z) \amp = z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\dots\\ \amp = \sum_{n=0}^{\infty} (-1)^n\frac{z^{2n+1}}{(2n+1)!} \amp\text{ valid}\;\forall z\nonumber\\ \cos(z) \amp = 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\dots\\ \amp = \sum_{n=0}^{\infty} (-1)^n\frac{z^{2n}}{(2n)!} \amp\text{ valid}\;\forall z\nonumber\\ e^z \amp = 1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{z^4}{4!} +\dots\\ \amp = \sum_{n=0}^{\infty} \frac{z^{n}}{n!} \amp\text{ valid}\;\forall z\nonumber\\ \ln(1+z) \amp = z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+\dots\\ \amp = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{z^{n}}{n} \amp\text{ valid for}\;\vert z\vert\lt 1\nonumber\\ (1+z)^p \amp = 1+p z+\frac{p(p-1)}{2!}\, z^2+\frac{p(p-1)(p-2)}{3!}\, z^3 +\dots\\ \amp = \sum_{n=0}^{\infty} \frac{p!}{n!(p-n)!}\, z^{n} \amp\text{ valid for}\;\vert z\vert\lt 1\nonumber \end{align*}

The last example, called a binomial expansion is valid even when \(p\) is not a positive integer. You may not know the meaning of \(p!\) in these cases, but it does have a definition. If necessary, just use the first line of the power series for \((1+z)^p\) instead of the second line.