With $a_0 =1$, $a_n = 8a_{n-1} + 10^{n-1}$
Let a generating function with it, $G(x) = \sum a_k x^k$ = $a_0 + \sum (8a_{k-1} +10^{k-1}) x^k =a_0 + \sum 8a_{k-1}x^k + \sum10^{k-1} x^k = a_0 + 8x\sum a_kx^k + x\sum10^{k} x^k $
Now I think that, $\sum 10^k x^k \neq\frac{1}{1-10x}$, since it has no limitation on $x$, such as $|x|<1$ (If so, $10^nx^n$ cannot be less than 1, too).
What should I do? How can I deal with $\sum 10^k x^k$ ??
Or is it fine to do just like that?