limit of sequence without general equation

I need to calculate the limit of the sequence :

$Cn = (1+a+a^2+...+a^n) / (1+b+b^2+...+b^n)$

$limit (1+a+a^2+...+a^n) / (1+b+b^2+...+b^n)$

I was thinking to try find the limit of $Cn+1 / Cn$, and based on the answer choose the limit.

but I got stuck here:

$(1+a+a^2+...+a^(n+1))*(1+b+b^2+...+b^n) / (1+b+b^2+...+b ^ (n+1))(1+a+a^2+...+a^n)$

Can anyone help pls?

1 Answer

Hint:

You can compute $C_n$ more explicitly in terms of $n$. For example, if $a\neq 1, b\neq 1,$ we can write $$C_n = \frac{\sum_{k=0}^n a^k}{\sum_{k=0}^n b^k} = \frac{1-a^{n+1}}{1-b^{n+1}}\frac{1-b}{1-a}.$$

• I don't understand how did you know that the sum of a^n and b^n is equal to this,can you explain ? Nov 9 '17 at 12:00
• Just compute $(1-a)\sum_{k=0}^na^k.$ If you don't like the symbol $\sum,$ you can write $(1-a)(1+a+a^2+...+ a^n)$ and see what happen. Nov 9 '17 at 13:06