# Series expansion: Taylor series takes huge amount of time

I'm working on a notebook, trying to expand the root of a cubic polynomial in Taylor series. When I type:

Series[(Sqrt[46656 a^2 - 864 (3 2^(1/3) a^(2/3) + 2 a B)^3] + 216 a)^(1/3) , {a, 0, 2}]


Mathematica takes an indefinite amount of time and I am forced to halt execution. After this occurs, even simple functions like Exp[x] will not compute and I have to restart the kernel.

Am I doing something wrong here? My computer is a month old, so I know the problem isn't old hardware.

• Works for me, but takes a bit more than two minutes on an i7-2820QM. – Yves Klett Nov 4 '13 at 17:56
• Do you really need an exact, symbolic result? It is likely to be huge so it may not be useful to you. You could convert the input to inexact numbers to a certain precision and work with that. – Szabolcs Nov 4 '13 at 18:00
• Series[N[(Sqrt[46656 a^2 - 864 (3 2^(1/3) a^(2/3) + 2 a bb)^3] + 216 a)^(1/3), 30], {a, 0, 2}] – Szabolcs Nov 4 '13 at 18:00
• You're not doing anything incorrect here. It seems that the Series code is using a fairly high order in some internal computations. I need to check whether there is solid reason for that, or whether it needs to be tamed to some extent. – Daniel Lichtblau Nov 4 '13 at 18:33
• If you take @Szabolcs advice: Series[(Sqrt[46656 a^2 - 864 (3 2^(1/3) a^(2/3) + 2 a B)^3] + 216 a)^(1/3) // N, {a, 0, 2}] works very quickly – Yves Klett Nov 4 '13 at 18:47

Assuming you're interested in the series expansion for positive values of the parameter a, you can use:
FullSimplify @ Series[

$$6 \sqrt[3]{a}+2 \sqrt[3]{2} \sqrt{a} \sqrt{-B}+\frac{2}{3} 2^{2/3} a^{2/3} B+\frac{2}{27} a^{5/6} (-B)^{3/2}-\frac{4}{81} a \left(\sqrt[3]{2} B^2\right)+\frac{5 a^{7/6} (-B)^{5/2}}{243 \sqrt[3]{2}}+\frac{8 a^{4/3} B^3}{2187}+\frac{35 a^{3/2} (-B)^{7/2}}{6561\ 2^{2/3}}+\frac{8\ 2^{2/3} a^{5/3} B^4}{19683}+\frac{2555 a^{11/6} (-B)^{9/2}}{4251528}-\frac{496 a^2 \left(\sqrt[3]{2} B^5\right)}{1594323}+O\left(a^{13/6}\right)$$