$ \lim_{n \to \infty}\Bigg(\dfrac{2 \cdot 10^n + 1 \cdot 10^{n-1} + 3 \cdot 10^{n-2} + 4 \cdot 10^{n-3}+...}{0\cdot 10^n + 1 \cdot 10^{n-1} + 1 \cdot 10^{n-2} + 2 \cdot 10^{n-3}+...}\Bigg) = 19 $
I noticed that the left-hand side converges to $19$, but I don’t know how to prove this identity, hence why I ask here for your help.
The generating function for the Fibonacci numbers is $$ f(x) = \sum_{n=0}^\infty F_n x^n = \frac{x}{1-x-x^2} $$ and the generating function for the Lucas numbers is $$ l(x) = \sum_{n=0}^\infty L_n x^n = \frac{2-x}{1-x-x^2} \, . $$ If we divide both numerator and denominator in your fraction by $10^n$ then the numerator converges to $l(1/10)$ and the denominator converges to $f(1/10)$. It follows that the limit is $$ \frac{l(1/10)}{f(1/20)} = \frac{2-1/10}{1/10} = 19 \, . $$