## installed_and_loaded.packages.
## prettydoc TRUEKey Idea 32 gives the n th term of the Taylor series of common funcOons. In Exercises 3 - 6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given funcOon and idenifying a pattern.
4. f(x) = sin x; c = 0
\[\begin{equation} \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n + 1)!} \end{equation}\] \[\begin{equation} f(x) = (-1)^0 \frac{x^{2 * 0 + 1}}{(2*0 + 1)!} + (-1)^1 \frac{x^{2*1+1}}{(2*1 + 1)!} + (-1)^2 \frac{x^{2*2+1}}{(2*2 + 1)!} + (-1)^3 \frac{x^{2*3+1}}{(2*3+1} + ... \end{equation}\] \[\begin{equation} f(x) = 1 * \frac{x^{1}}{(1)!} + -1 * \frac{x^{3}}{(3)!} + 1 * \frac{x^{5}}{(5)!} + -1 * \frac{x^{7}}{(7)!} + ... \end{equation}\] \[\begin{equation} f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \end{equation}\]