I'm currently learning the prove of sum of geometric series on Khan Academy.
I understand the behaviour of the function when $|r| > 1$, when $|r| < 1$, when $r = 0$ and when $r = -1$, but I am a bit confused by its behaviour when $r = 1$.
The narrator said that when $r = 1$, the limit function is undefined because the denominator of the limit function would be $0$, and the behaviour of the limit function is UNDEFINED, which I do understand.
My confusion arises when I tried to substitute $r = 1$ into the original function for sum of geometric series,
if $r = 1$, then every term would equal to a, and the sum of the geometric series would approach infinity, so its behaviour is DEFINED.
So when $r = 1$, behaviour of sum function is DEFINED, but behaviour of limit function is UNDEFINED, but sum function also equal to limit function?!
This is causing me so much confusion.