Appendix C Power Series of Some Common Analytic Functions
We list the power series representation about 0 of some common analytic functions and indicate their intervals of convergence.
\begin{align*}
\frac{1}{1-x} & = \sum_{n=0}^{\infty} x^n = 1 + x+ x^2 + x^3 + \cdots
\quad (-1 \lt x \lt 1)\\
\ln(1+x) & = \sum_{n=0}^{\infty} (-1)^n \frac{x^{n+1}}{n+1} = x -
\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (-1 \lt x
\le 1)\\
\arctan(x) & = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{2n+1}
= x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \quad
(-1 \le x \le 1) \\
e^x & = \sum_{n=0}^{\infty} \frac{x^n}{n!} =1 + x + \frac{x^2}{2!} +
\frac{x^3}{3!} + \cdots \quad (-\infty \lt x \lt \infty) \\
\sin(x) & = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}
= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \quad
(-\infty \lt x \lt \infty)\\
\cos(x) & = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 -
\frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \quad
(-\infty \lt x \lt \infty)
\end{align*}
Hit the "Evaluate" button below, you will see the function and its \(k\) th Taylor polynomial centered at the origin. Get an idea of how these approximations look like by varying the parameters.
