Taylor Series - crash-course
Outcome • Remember the general Taylor formula and the Maclaurin special case • Know how to grab the first n terms for quick approximations • Recall the remainder term’s shape to bound your error Definition For a function $f(x)$ with enough derivatives near $a$: $$ f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^{k} $$ • If $a=0$ it’s called a Maclaurin series. The $n^{\text{th}}$-degree Taylor polynomial: $$ P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^{k} $$ Remainder / error after $n$ terms (Lagrange form): ...