This is probably easy, however I can't figure out... This is an introductory problem for C Bender's book.
The equation is $x^3 - (4 + \epsilon)x + 2 \epsilon = 0$. By assuming the solution is in the format of $\sum a_n \epsilon^n$, and collect items of powers of $\epsilon$, we can work out the coefficients one by one.
Here are the equations I got:
$\epsilon^0: a_0^3 - 4a_0 = 0$
$\epsilon^1: 3a_0^2a_1 - 4a_1-4a_0 + 2 = 0$.
$\epsilon^n: \sum_{i+j+k = n}a_ia_ja_k - 4a_n -a_{n-1} = 0$.
Question: How to show the series converge for $|\epsilon|<1$?