I have a geometric sequence whose n-th term is $11^n$:
$1,11,121,1331,...$
I want to know if the pattern continues forever. I understand that if I sum these numbers up, my common ratio is 11 which is not between -1 and 1 so the sum doesn't converge but I don't see any reason why this pattern would fail to terminate since there are countably infinite multiples of 11.
The finite sum $$ 1 + x + x^2 + \cdots + x^n = \frac{x^{n+1} -1 }{x-1} $$ whenever $ x \ne 1$.
So you can substitute $11$ for $x$ for any particular $n$ you like.
You can only sum the series "forever" when $-1 \le x < 1$.