I happen to read a book, in which it states that,
if $P_n(x)$ is the $n+1$ th Taylor polynomial (that is, $P_n(x)=a_0+a_1x+\cdots+a_{n}x^n$) of the function $\sqrt{x+a}$ (a>0)at $x=0$, then there is a polynomial $Q_n(x)$ such that
$P_n^2(x)=x+a+x^{n+1}Q_n(x).$
I have tried to calculate for small $n$, it shows the claim is right.
But I cannot find an easy way to prove it, or understand the result clearly. And is there a more general result for other functions, like $\sqrt[3]{x+a}$?