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?



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

  • $\begingroup$ I don't understand how did you know that the sum of a^n and b^n is equal to this,can you explain ? $\endgroup$
    – kal pola
    Nov 9 '17 at 12:00
  • $\begingroup$ 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. $\endgroup$
    – A. PI
    Nov 9 '17 at 13:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.