<s>[INST] 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?
 [/INST] 
$$f(x) =\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$

The argument of the square root is 0, hence $f$ has no derivative in 0. That is why you get non integer powers.
First, get the equivalent:
$$\sqrt{1+8 x}-1 \sim 4x\\
f(x) = \frac 12\sqrt{-1+\sqrt{1+8 x}}\sim \frac 12\sqrt{4x} = \sqrt{x}
$$
and then:
$$
f(x) -\sqrt{x} 
= \frac 12\sqrt{-1+\sqrt{1+8 x}} - \frac 12\sqrt{4x}
\\= \frac 12 \frac{-1+\sqrt{1+8 x} +4x}
{\sqrt{-1+\sqrt{1+8 x}} - \sqrt{4x}}
\sim 
\frac 12 \frac{-\frac 18 (8x)^2}{4\sqrt{x}}=-x\sqrt{x}
$$
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