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.
 A: 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.
A: We recall the Ratio Test: Let $\sum\limits_{k=0}^{\infty} b_k$ be a series. If $\lim_{k \to \infty} \vert \dfrac{b_{k+1}}{b_k} \vert = L < 1$ then the series converges. If $L >1$, then the series diverges; if $L=1$ then we cannot say anything for certain.
I will show how this works for $\sum\limits_{k=0}^{\infty} k a^k$. To setup the test, we examine $$\lim_{k \to \infty} \vert \dfrac{b_{k+1}}{b_k} \vert  = \lim_{k \to \infty} \vert \dfrac{(k+1)a^{k+1}}{k a^k} \vert $$ which we may rewrite as $$\lim_{k \to \infty} \vert \dfrac{k+1}{k} a \vert  = |a|.$$
Therefore, the series converges whenever $|a| < 1$.
