Discussion Week 14

Chapter 8.8, # 27

Find the Taylor Series for $f(x)=sin(2x+3)$.

From Key Idea 8.8.1, and Theorem 8.8.2, if we let $f(x) = sin(x)$ and $h(x) = 2x+3$, then

$sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \cdot x^{2n+1}$

Therefore,

$$\begin{align*} f(h(x)) & = sin(2x+3) \\ & = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \cdot (2x+3)^n \end{align*}$$

The first few terms are,

$1 - \frac{1}{3!}(2x+3) + \frac{1}{5!}(2x+3) - \frac{1}{7!}(2x+3) + \dots$