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