Series expansion involving fractional powers

Question

I try to analyze a behavior of the function $\sqrt{1 - \frac{x}{(1 + \sqrt{x})^{2}}}$ around 0.

The Taylor expansion does not work, as the second derivative at 0 diverges. By calculating the series with Mathematica I get however $\sqrt{1 - \frac{x}{(1 + \sqrt{x})^{2}}} \approx 1-\frac{x}{2}+x^{\frac{3}{2}}+...$ .

How one gets such a series expansion involving fractional powers?

Answer 1

With the substitution $z=\sqrt x$, the function becomes $$f(z):=(1+2z)^{1/2}(1+z)^{-1}.$$

Then using the Leibniz rule for the differentiation of a product,

$$f^{(n)}(z)=(-1)^n\sum_{k=0}^n s_k\binom nk\frac12\frac12\frac32\cdots\frac{2k-1}22^k(1+2z)^{1/2-k}(n-k)!(1+z)^{-1-n+k}$$

and

$$\frac{f^{(n)}(0)}{n!}=(-1)^n\sum_{k=0}^n s_k\frac{(2k-1)!!}{k!}$$

where the sign $s_k$ is $1$ for $k=0$ and $-1$ otherwise (we let $(-1)!!=1$). These coefficients can be computed incrementally.

This gives us the Taylor development in terms of $z$, which is $x^{1/2}$.