How to find limit of a sequence $\lim_{n\to \infty} \frac{1+a+a^2+...+a^n}{1+b+b^2+...+b^n}$? I have no idea how to solve this: $$\lim_{n\to \infty} \frac{1+a+a^2+...+a^n}{1+b+b^2+...+b^n}$$ for: $|a|<1$, $|b|<1 $
I would be happy for any advice.
Thank you

The limit is $$\lim_{n\to \infty}\left(\frac{1+a+a^2+...+a^n}{1+b+b^2+...+b^n}\cdot\frac{1-a}{1-b}\cdot\frac{1-b}{1-a}\right)=\lim_{n\to \infty} \frac{1-a^{n+1}}{1-b^{n+1}}\cdot\frac{1-b}{1-a}=\frac{1-b}{1-a}$$
EDIT
My thinking (pedagogic insert):
$1+a+a^2+...+a^n\;$ is the sum of first $n+1$ terms of geometric series.
The calculation that simplifies the sum and helps when looking for the limit, is multiplying by  $\frac{1-a}{1-a}.$
Similarly for $\;1+b+b^2+...+b^n.$
Both numerator and denominator converge (since  $|a|,|b|<1$).  
