Answer
Absolutely convergent
Work Step by Step
Given $$\sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k}$$
By using the Ratio Test, since $a_k=k\left(\dfrac{2}{3}\right)^{k}$ and $a_{k+1}=(k+1)\left(\dfrac{2}{3}\right)^{k+1}$
\begin{align*}
\lim _{k \rightarrow \infty}\left|\frac{a_{k+1}}{a_{k}}\right|&=\lim _{k \rightarrow \infty}\left|\frac{(k+1)\left(\frac{2}{3}\right)^{k+1}}{k\left(\frac{2}{3}\right)^{k}}\right|\\
&=\left(\frac{2}{3}\right)\lim _{k \rightarrow \infty} \frac{k+1}{k} \\
&=\frac{2}{3} \lim _{k \rightarrow \infty}\left(1+\frac{1}{k}\right)\\
&=\frac{2}{3}(1)=\frac{2}{3}<1
\end{align*}
Then $\displaystyle\sum_{n=1}^{\infty} k\left(\frac{2}{3}\right)^{k}$ is absolutely convergent