All Questions

Consider a function $$ f:\mathbb{R}^n\rightarrow\mathbb{R}^m $$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$. Does there ...

Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$. Expand the function $\frac1f$ at $\infty$ so that $$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$ Does it follow that each $...

I'm in trouble understanding this theorem: For a given function, differentiable n times at a given point $x_0$, there exist a unique polynomial $P_n$ (of degree $\le$n) such that $$\forall \; k=0,...,...