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{eqnarray} \sin(z) &=& \displaystyle\sum_{n=0}^{\infty} (-1)^n\frac{z^{2n+1}}{(2n+1)!} \qquad\qquad\qquad\qquad \hbox{valid $\;\forall z$}\\ &=& z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\dots\\ \end{eqnarray} \begin{eqnarray} \cos(z) &=& \displaystyle\sum_{n=0}^{\infty} (-1)^n\frac{z^{2n}}{(2n)!} \qquad\qquad\qquad\qquad\qquad \hbox{valid $\;\forall z$}\\ &=& 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\dots\\ \end{eqnarray} \begin{eqnarray} e^z &=& \displaystyle\sum_{n=0}^{\infty} \frac{z^{n}}{n!} \qquad\qquad\qquad\qquad\qquad\qquad \hbox{valid $\;\forall z$}\\ &=& 1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{z^4}{4!}+\dots\\ \end{eqnarray} \begin{eqnarray} \ln(1+z) &=& \displaystyle\sum_{n=1}^{\infty} (-1)^{n+1}\frac{z^{n}}{n} \qquad\qquad\qquad\qquad \hbox{valid for $\;\vert z\vert<1$}\\ &=& z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+\dots\\ \end{eqnarray} \begin{eqnarray} (1+z)^p &=& \displaystyle\sum_{n=0}^{\infty} \frac{p!}{n!(p-n)!}\, z^{n} \qquad\qquad\qquad\qquad \hbox{valid for $\;\vert z\vert<1$}\\ &=& 1+p z+\frac{p(p-1)}{2!}\, z^2+\frac{p(p-1)(p-2)}{3!}\, z^3+\dots \end{eqnarray} You may not know the meaning of $p!$ if $p$ is not a positive integer. If necessary, just use the second line of the power series for $(1+z)^p$ instead of the first line. (The factorial function can, in fact, be extended to be a valid function on the domain of all complex numbers, except the negative integers; the process is called analytic continuation. If you are curious, look up “gamma function” in any good mathematical methods text.)