**Question:**

For the cubic equation

$$x^3+x-\varepsilon = 0 \qquad \varepsilon \rightarrow 0$$

find the first three terms of the asymptotic expansion of the solutions.

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**Attempt:**

I observed that a solution is $x=0$ when $\varepsilon = 0$, so I picked the starting point $x\_0=0$.

I then rearranged the equation as $x = (\varepsilon - x)^{1/3}$, to give the recurrence relation

$$x\_{n+1} = (\varepsilon - x\_n)^{1/3}$$

Plugging it in, we get

\begin{align}
x\_0 & = 0 \\
x\_1 & = \varepsilon^{1/3} \\
x\_2 & = (\varepsilon - \varepsilon^{1/3})^{1/3} \sim \mathcal O(\varepsilon^{1/9})
\end{align}

and I am confused, because apparently future iterations are supposed to match previous iterations at the leading orders.

But in this case, I had $x\_1 = \mathcal O(\varepsilon^{1/3})$ and $x\_2 = \mathcal O(\varepsilon^{1/9})$.

What have I done wrong? Any hints are much appreciated. Please help T\_T