Find complicated Taylor Series According to some software, the power series of the expression,
$$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$
around $x=0$ is
$$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$
When I try to do it I find that I can't calculate Taylor because there are divisions by zero. Also I do not understand how Taylor could give non integer powers.
Does anybody know how this expression is calculated?

Close to $x=0$,$$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}} \simeq \frac{1}{2} \sqrt{-1+ (1+4x)}\simeq \sqrt x$$ So, this must be the start of the development (in order that, locally, your expansion looks like the formula)
If you start using $$\sqrt{1+8 x}  \simeq 1+4 x-8 x^2+32 x^3+O\left(x^4\right)$$ then $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}} \simeq \frac{1}{2}\sqrt {4 x-8 x^2+32 x^3+O\left(x^4\right)}=\sqrt {x-2 x^2+8 x^3+O\left(x^4\right)}$$ Now, extract $\sqrt x$ and you get $$\sqrt x \sqrt {1-2x+8 x^2}$$ Develop the second square root and you arrive to your wanted result.  
Suppose we change the problem to $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 \sqrt{x}}}$$ For the same reasons, the first term should be $x^{1/4}$ and the development would be $$x^{1/4}-x^{3/4}+\frac{7 x^{5/4}}{2}-\frac{33 x^{7/4}}{2}+O\left(x^{9/4}\right)$$
