We already know the power series for $\frac{1}{1+x}$, $\ln(1+x)$, $\frac{1}{(1-x)^2}$, and $\tan^{-1}(x)$. Using the formula from the last slide, $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n,$$ we derive the Taylor series for a number of other common functions:

  • $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

  • $\displaystyle e^{2x} = \sum_{n=1}^\infty \frac{2^n}{n!} x^n$

  • $\displaystyle e^{x^2} = \sum_{n=0}^\infty \frac{x^{2n}}{n!}$

  • $\displaystyle\sin(x) = \sum_{n=0}^\infty \frac{(-1)^{n} x^{2n+1}}{(2n+1)!} =x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $

  • $\displaystyle\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots $

  • $\displaystyle\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} + \cdots $