Taylor series for $\sqrt{x}$? I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particular reason no one shows Taylor series for exactly $\sqrt{x}$?

I assume you are talking about the Taylor series at $0$ for $\sqrt{x}$. Let's try to compute the Taylor series at $0$:
$$
f(x)=f(0)+f'(0)(x-0)+f''(0)\frac{(x-0)^2}2+\dots
$$
$f(0)=0$, but $f'(x)=\frac1{2\sqrt{x}}$ blows up at $x=0$. Since $\sqrt{x}$ doesn't have a first derivative at $0$, it doesn't have a Taylor series there.
