As we know, Taylor's formula is a method which allows us to find power series for specific functions in a systematical way. I am trying to understand it, not just apply it. Let $f(x)$ be the power series:
$$f(x)=\sum_{i=0}^{\infty} a_{i}(x-x_{0})^i$$
My book states when $f(x)$ is convergent for $(x-x_{0})$ small enough, we can find the coefficient $a_{0}$ by setting $x=x_{0}$. Therefore:
$$f(x_{0})=\sum_{i=0}^{\infty} a_{i}(x_{0}-x_{0})^i= a_{0}$$
But, would not it be the indetermination $0^0$? How could we solve this equation then?
Thank you
$0^0$ can be seen as
$$\lim_{x\to 0^+}x^x=\lim_{x\to0^+}e^{x\ln (x)}=e^0=1$$