$\displaystyle \displaystyle \begin{align*} \frac{1}{x^2 + x + 1} &= \frac{1}{1 - \left(-x^2 - x\right)}\textrm{ which is the closed form of a geometric series} \\ &= \sum_{n = 0}^{\infty}(-x^2 - x)^n\textrm{ provided }|-x^2 - x| < 1 \end{align*}$
Just a little precisation: it was requested the Taylor expansion of $\displaystyle f(x)=\frac{1}{1+x+x^{2}}$ around x=0 [in that case we properly have a McLaurin extension...], so that the result must be in the form...
$\displaystyle \frac{1}{1+x+x^{2}} = \sum_{n=0}^{\infty} a_{n}\ x^{n}$ (1)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
$\displaystyle f(x) = \frac{1}{x^2+x+1} = \frac{1}{x^2+x+1}\cdot \frac{1-x}{1-x}$
$\displaystyle = \frac{1-x}{1-x^3} = (1-x)(1+x^3 +x^6+x^9+\dots)$
So if
$\displaystyle f(x) = \sum_{n=0}^{\infty} c_n x^n$
then
$\displaystyle c_{18}=1$
$\displaystyle c_{19}=-1$
$\displaystyle c_{20}=0$
I started with the right way but soon I lost it!... In terms of Zeta-Tranform, difference equation corresponding to the function $\displaystyle h(z)=\frac{1}{1+z+z^{2}}$ is...
$\displaystyle a_{n}+a_{n-1}+a_{n-2}= \delta_{n}$ (1)
... where...
$\displaystyle \delta_{n} =\begin{cases}1 &\text{if }n=0\\ 0 &\text{if } n>0\end{cases} $ (2)
... so that the coefficients of the Taylor series are found symply expanding (1)...
$\displaystyle a_{0}= 1 - a_{-1}-a_{-2}= 1$
$\displaystyle a_{1}= 0 - a_{0}-a_{-1}= -1$
$\displaystyle a_{2}= 0 - a_{1}-a_{0}= 0$
$\displaystyle a_{3}= 0 - a_{2}-a_{1}= 1$
$\displaystyle a_{4}= 0 - a_{3}-a_{2}= -1$
... so that is...
$\displaystyle \frac{1}{1+z+z^{2}}= 1 - z + z^{3} - z^{4} + z^{6}-z^{7}+...$ (3)
... as awkward found...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
From your original post:
$\displaystyle f(x) = \sum_{n=0}^{\infty} c_n x^n$
So $\displaystyle c_n$ is the coefficient of $\displaystyle x^n$ in the series for f(x).
But the problem did not ask for a general form of the coefficient, it simply asked for the value of
$\displaystyle c_{18} + c_{19} + c_{20}$,
which can be found without necessarily finding the general coefficient.