Taylor Polynomials and Series

Suppose $f$ is $n$ times differentiable on $I$. Then, the nth Taylor polynomial of $f$ at $x=a\in I$ is

$$T_n(f(x);a)=\sum _{j=0}^n\frac{f^{(j)}(a)}{j!}(x-a{)}^j.$$

Taylor polynomials are good approximations for functions.

Useful Taylor Polynomials §

[!math|{“type”:“lemma”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:0}] Lemma 1. If $f(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_m{x}^m$, then $f$ is infinitely differentiable and

T_{n}(f(x);0)=\begin{cases} a_{0}+a_{1}x+\dots+a_{n}x^{n} \quad \text{if } n<m \\ a_{0}+a_{1}x+\dots+a_{m}x^{m} \quad \text{if } n\geq m

\end{cases}

We can prove the above using induction. ## Sin and Cos

T_{n}(\sin x;0)=\sum_{j=1}^{n} (-1)^{j+1}\frac{x^{2j+1}}{(2j+1)!}

We can once again prove the above using induction on the derivative of $\sin$.