What kind of functions cannot be described by the Taylor series? Why is this? |
您所在的位置:网站首页 › infinitely differentiable › What kind of functions cannot be described by the Taylor series? Why is this? |
We have the somewhat famous function: $$f(x)=\begin{cases}e^{-1/x^2}&x\neq 0\\ 0&x=0 \end{cases}$$ is infinitely differentiable at $0$ with $f^{(n)}(0)=0$ for all $n$, so, even though the function is infinitely differentiable, the Taylor series around $0$ does not converge to the value of the function for any $x>0$. Technically, any function that is infinitely differentiable at $a$ has a Taylor series at $a$. Whether you find that Taylor series useful depends on what you want the series to do. For example, if given a $g$ infinitely differentiable at $0$, the we know that there exists $C,\epsilon>0$ such that: $$\left|g(x)-\sum_{k=0}^{n} \frac{g^{(k)}(0)}{k!}x^k\right| |
CopyRight 2018-2019 办公设备维修网 版权所有 豫ICP备15022753号-3 |