# Jared Antrobus @ University of Kentucky

Email me at jantrobus@uky.edu.
My office is in Patterson Office Tower 722.
My mailbox is in POT 715.

jantrobus@uky.edu
Office in POT 722
Mailbox in POT 715

This semester, I can be found Monday 10am to 11am in the Mathskeller. I am available to meet at other times throughout the week by appointment.

> Home / squirrel / 2015 / Fall / MA114

## MA 114 Calculus 2

### Taylor Polynomials for Sine

The Taylor series expansion for $\sin(x)$ is $$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1},$$ which converges on $(-\infty,\infty)$. Notice that only the odd terms are nonzero. The first few Maclaurin polynomials (Taylor polynomials centered at $x=0$) for $\sin(x)$ are: \begin{align} T_1(x)&=x\\ T_3(x)&=x-\frac{x^3}{6}\\ T_5(x)&=x-\frac{x^3}{6}+\frac{x^5}{120}\\ T_7(x)&=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}\\ T_9(x)&=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\frac{x^9}{362880}\\ \end{align} This is reflected in the interactive diagram below. As the red Taylor polynomial increases in degree, it becomes a better approximation for the sine function. Notice how the graph does not change on the even terms. You can also drag the point at $(0,0)$ left and right to change the center, but keep in mind that the resulting graphs correspond to different polynomials than those above.