I hope title is understandable, wasn't sure on how to translate this task from my language.
Below is my solution to this problem, is logic behind it correct?
$$A(z)=\sum_{n=0}^{\infty}a_nz^n=1+\frac{5}{2}z+3z^2+\sum_{n\geq3}^{\infty}a_nz^n$$ $$=1+\frac{5}{2}z+3z^2+2^3+\sum_{n\geq3}^{\infty}(2z)^n$$ $$=1+\frac{5}{2}z+3z^2+2^3+[(2z)^3+(2z)^4+(2z)^5...]$$
$$b_n=(2_z)^n$$
$$\frac{b_{n+1}}{b_n}=2z$$
$$q=2z$$ $$W_0=(2z)^3$$ $$|2z|<1|$$ $$=1+\frac{5}{2}z+3z^2+2^3+\frac{(2z)^3}{1-2z}$$
I need to make sure on test that I didn't make mistakes so if its correct, could any of you explain on how to check if answer is right?
I think your work is wrong: $$\begin{align} A_n(x)&=\sum_{n \in \mathbb {N}}a_n x^n\\ &=a_0x^0+a_1x^1+a_2x^2+\sum_{n \geqslant3}a_n x^n\\ &=19+x+a_2x^2+\sum_{n \geqslant 3}2^{n+3}x^n\\ &=19+x+a_2x^2+2^3\sum_{n \geqslant 3}(2x)^n\\ \end{align}$$