This is an offshoot of the question: http://mathematica.stackexchange.com/questions/132635/series-expansions-in-mathematica. In that question, I thought I would simplify my problem and ask the simplest version of it. While I have gotten that answered, it seems there is still an issue with the real Mathematica code that I have. 

I'm studying the Hurwitz Lerch Transcendant $\Phi(z,s,k)$ and am particularly interested only with $\Phi(z,n,\frac{1}{2})$ for $n\in {\mathbb Z}$. This function satisfies the following property. Define

    f[z_, n_] :=  2^(n - 1)/Sqrt[z] (PolyLog[n, Sqrt[z]] - PolyLog[n, -Sqrt[z]])

Then,

    HurwitzLerchPhi[z, n, 1/2] == f[z, n]

Now, Mathematica knows this fact. For instance, if I write

    Plot[(HurwitzLerchPhi[z, n, 1/2] - f[z, n]) /. n -> 9, {z, 0.0000001,1}]
    Plot[(HurwitzLerchPhi[z, n, 1/2] - f[z, n]) /. n -> 13, {z, 0.0000001,1}]
    Plot[(HurwitzLerchPhi[z, n, 1/2] - f[z, n]) /. n -> 21, {z, 0.0000001,1}]
    Plot[(HurwitzLerchPhi[z, n, 1/2] - f[z, n]) /. n -> 3, {z, 0.0000001,1}]

the plots are identically 0. I chose some random integers above. You can check this with other ones as well. We can also check this numerically with arbitrary accuracy. 

However, if I consider the following code into Mathematica

    FullSimplify[Series[HurwitzLerchPhi[1 - Sqrt[z], 3, 1/2] - f[1 - Sqrt[z], 3], {z, 0, 6}], 0 < z < 1]

I get 
$$
\frac{z^2}{4}+\frac{3 z^{5/2}}{8}+\frac{55 z^3}{96}+\frac{131 z^{7/2}}{192}+\frac{19219 z^4}{23040} +\frac{42493
   z^{9/2}}{46080}\\\qquad \qquad \qquad \qquad +\frac{268843 z^5}{258048}+\frac{957181 z^{11/2}}{860160}+\frac{107031761
   z^6}{88473600}+O\left(z^{13/2}\right)
$$
which is demonstrably non-vanishing. The problem seems to occur if the argument is $1-\sqrt{z}$. For instance, there is no issue if I take the argument to be $1-z$ or $\sqrt{1-z}$, but you do get a non-zero answer with $\sqrt{1-\sqrt{z}}$.


I don't understand at all what Mathematica is doing here. Further, the issue does not seem to be one of convergence (As suggested in the answer to http://mathematica.stackexchange.com/questions/132635/series-expansions-in-mathematica). 

What is going on?