So I'm doing exercises from Dasgupta's Algorithms. The exercise i'm having trouble with is:
Show that, if $c$ is a positive real number, then $g(n) = 1 + c + c^2 +...+c^n$ is:
$\Theta(1)$ if $c<1$
$\Theta(n)$ if $c=1$
$\Theta(c^n)$ if $c>1$
(I dont know if this is a hint but it is included in the text: "The moral: in big-$\Theta$ terms, the sum of a geometric series is simply the first term if the series is strictly decreasing, the last term if the series is strictly indreasing or the number of terms if the series in unchanging")
The only one that makes sense for me is 2) where $1+1^2+..+1^n$ is the same as $n+1$, and removing the 1 gives $O(n)$. I dont know if my reasoning makes sense, but thats all i've got. I have no idea where to start or how to think on the other two. Any suggestions?