Convergence demonstration? 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.

The first sum converges because it contains finitely many finite terms. You can use the ratio test for proving the convergence of the others as follows:
\begin{align}
\lim_{k \rightarrow \infty} {\Big|}\frac{a^{k+1}}{a^k}{\Big|} &= \lim_{k \rightarrow \infty} |a| < 1 \\
\lim_{k \rightarrow \infty} {\Big|}\frac{(k+1)a^{k+1}}{ka^k}{\Big|} &= \lim_{k \rightarrow \infty} {\Big|}{\Big(}\frac{1}{k}+1{\Big)}a{\Big|} = |a| < 1 \\
\lim_{k \rightarrow \infty} {\Big|}\frac{(k+1)ka^{k+1}}{k(k-1)a^k}{\Big|} &= \lim_{k \rightarrow \infty} {\Big|}{\Big(}\frac{1+\frac{1}{k}}{1-\frac{1}{k}}{\Big)}a{\Big|} = |a| < 1 \\
\lim_{k \rightarrow \infty} {\Big|}\frac{(k+1)^2a^{k+1}}{k^2a^k}{\Big|} &= \lim_{k \rightarrow \infty} {\Big|}{\Big(}1+\frac{1}{k}{\Big)}^2a{\Big|} = |a| < 1 \\
\end{align}
In all the above cases, the ratio limit is $< 1$. Hence all series converge absolutely.
