I got the following assigned as homework: "Demonstrate that the following series are convergent:"
$\sum_{k=0}^N a^k\\ \sum_{k=0}^\infty a^k \\ \sum_{k=0}^\infty ka^k \\\sum_{k=0}^\infty k(k-1)a^k\\ \sum_{k=0}^\infty k^2a^k$
I know most of these converge when $|a| < 1$, but I'm not sure how I'm supposed to prove this, cause I don't think the ratio test applies with all the series? Especially the first one.
I apologize if this a really stupid question, cause it feels like one, but this is the first time I'm dealing with the convergence subject.