Finding the limit of $( 1 + a + a^2 + \ldots+ a^n)/(1 + b + b^2 + \ldots+ b^n)$ I have some problems finding the limit of $$\frac{ 1 + a + a^2 +\cdots + a^n}{1 + b + b^2 + \cdots + b^n}.$$
$0\le a,b \le +∞$
Here is what I got :

Forcefuly factorize $a^n$ and $b^n$ :
     $$ \frac{a^n ( 1 + \frac1{a} + \frac1{a^2} + ... + \frac1{a^n})}{b^n( 1 + \frac1{b} + ... + \frac1{b^n})} 
      = \left(\frac ab\right)^n\cdot \frac{\dfrac{1-\frac{1}a^{n+1}}{1-\frac{1}a}}{\dfrac{1-\frac{1}b^{n+1}}{1-\frac{1}b}} $$

From here I'm kinda stuck since I don't know the limit of $\left(\frac ab\right)^n$ unless I take all the possible cases. Same goes for the other part.
I want to know the next step into solving this limit.

$\displaystyle\lim_{n\rightarrow\infty}\frac{1+a+\cdots+a^n}{1+b+\cdots+b^n}=\displaystyle\lim_{n\rightarrow\infty}\frac{1-b}{1-a}.\frac{1-a^{n+1}}{1-b^{n+1}}=\frac{1-b}{1-a}$ if $1<a\leq b$. 
