Finite sum of expression as exponent goes to infinity I am trying to understand the limit below:
$\lim_{m \to \infty }\sum_{i=1}^{n} r_i^{m+1}$ while $|r_d| \leq |r_{d-1}| \leq \dots \leq |r_2| \leq |r_1| \leq 1$. This means that the range of $r_1$ include the ranges of the rest $r_k$ .Could you please help to identify which rules apply to compute such a limit?
Thank you in advance!

If $|r_i| < 1$ then $\lim_{m \to \infty} r_i^{m+1} = 0$. Further, if $r_i = -1$ then $\lim_{m \to \infty} r_i^{m+1}$ does not exist. Thus if $-1 < r_i \leq 1$ for all $i = 1,2,\ldots,n$, then
$$
\lim_{m \to \infty} \sum_{i=1}^{n} r_i^{m+1}
$$
equals the number of indices $i$ for which $r_i = 1$. If there is any index $i$ for which $r_i = -1$, then the limit does not exist.
