Series Expansions in Mathematica

Question

Bug introduced in 5.0 or earlier and fixed in 11.1.1

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Define variable z with the following assumptions

$Assumptions = 0 < z < 1

I put in the following two codes in Mathematica

t1 = Simplify[Series[ArcTanh[(1 - Sqrt[z])^2], {z, 0, 3}]]

t2 = Simplify[Series[ArcTanh[(1 - Sqrt[z])^2] // TrigToExp, {z, 0, 3}]]

The codes are identical to each other except for the extra TrigToExp. Yet, the outputs are completely different

$$t_1 = -\frac{\log (z)}{4}-\frac{\sqrt{z}}{4}+\frac{z}{16}+\frac{5 z^{3/2}}{48}+\frac{z^2}{16}+\frac{z^{5/2}}{40}+O\left(z^{7/2}\right) ~~~~~~~~~~~~~~~\\ t_2 = -\frac{\log (z)}{4}-\frac{\sqrt{z}}{4}+\frac{z}{16}+\frac{5 z^{3/2}}{48}+\frac{9 z^2}{128}+\frac{9 z^{5/2}}{320}+\frac{z^3}{768}+O\left(z^{7/2}\right) $$

Why are the two results different and which result should I trust?

Answer 1

The which one can you trust, can be answered by plotting the error,

$$\left |\tanh ^{-1}\left(\left(1-\sqrt{z}\right)^2\right)-\text{t1} \right | \text{ and } \left |\tanh ^{-1}\left(\left(1-\sqrt{z}\right)^2\right)-\text{t2} \right |$$

t1 = Simplify[Series[ArcTanh[(1 - Sqrt[z])^2], {z, 0, 3}]] //Normal;

t2 = Simplify[Series[ArcTanh[(1 - Sqrt[z])^2] // TrigToExp, {z, 0, 3}]] //Normal;

Plot[{Abs[ArcTanh[(1 - Sqrt[z])^2] - t1], 
  Abs[ArcTanh[(1 - Sqrt[z])^2] - t2]}, {z, 0, 1}, 
 PlotStyle -> {Blue, Green}, PlotLabels -> {"f(z)-t1", "f(z)-t2"}]

Looks like t2 would be my choice for 0 < z < 1 / 2 and t1 for the rest.