Find the power series representation of the function I have to find $a_n$ so, my first attempt was to do partial fraction, but with no real solution. Any recommendation will be appreciate  $$\frac{1}{(2x-3)(x^2-x+1)}=\sum_{n=0}^\infty a_n (x-1)^n$$
 A: Let us replace $x$ with $z+1$ for the sake of simplicity. We have to find the sequence $\{a_n\}_{n\geq 0}$ given by:
$$ f(z)=\frac{1}{(2z-1)(z^2+z+1)}=\sum_{n\geq 0}a_n z^n \tag{1} $$
but $f(z)$ is a meromorphic function with simple poles at $z=\frac{1}{2},\,z=\omega,\,z=\omega^2$, where $\omega=e^{\frac{2\pi i}{3}}$.
By computing the residues of $f(z)$ at such points we get a partial fraction decomposition for $f(z)$:
$$ f(z) = \frac{2}{7}\cdot \frac{1}{x-\frac{1}{2}}-\frac{1}{7}\cdot\frac{3-x-2x^2}{1-x^3} \tag{2}$$
and by performing expansions as geometric series it follows that:
$$ a_n = -\frac{4}{7} 2^n-\frac{1}{7}\cdot\left\{\begin{array}{rcl}3 & \text{if} & n\equiv 0\pmod{3}\\ -1 & \text{if} & n\equiv 1\pmod{3}\\ -2 & \text{if} & n\equiv 2\pmod{3}\end{array}\right.\tag{3}$$
that can be stated as: 
$$ a_n \text{ is the closest integer to } -\frac{2^{n+2}}{7}. \tag{4}$$
A: Note that the geometric series is valid also in the complex field
$$
\frac{1}
{{1 - z}} = \sum\limits_{0\, \leqslant \,n} {z^{\,n} } \quad \left| \begin{gathered}
  \;z \in \;\mathbb{C}\; \hfill \\
  \,\left| z \right| < 1 \hfill \\ 
\end{gathered}  \right.
$$
and that the partial fraction decomposition can be operated also in the complex field,
so that in your case we can write
$$
\begin{gathered}
  f(z,\omega ) = \frac{1}
{{\left( {2z - 3} \right)\left( {z - \omega } \right)\left( {z + \omega } \right)}} =  \hfill \\
   = \frac{1}
{{2\left( {z - 3/2} \right)\left( {z - \omega } \right)\left( {z + \omega } \right)}} =  \hfill \\
   =  - \frac{1}
{{3\left( {3/2 - \omega } \right)\left( {3/2 + \omega } \right)\left( {1 - 2/3\,z} \right)}} + \frac{1}
{{4\omega ^2 \left( {3/2 - \omega } \right)\left( {1 - z/\omega } \right)}} + \frac{1}
{{4\omega ^2 \left( {3/2 + \omega } \right)\left( {1 - \left( { - z/\omega } \right)} \right)}} \hfill \\ 
\end{gathered} 
$$
and the expression for $a_n$ follows easily.
