Answer
Converges
Work Step by Step
Apply the ratio test.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty} \dfrac{(k+1)(\dfrac{1}{2})^{k+1}}{k(\dfrac{1}{2})^k} \\=\dfrac{1}{2} \lim\limits_{k \to \infty} (\dfrac{k+1}{k}) \\=\dfrac{1}{2} \times \lim\limits_{k \to \infty} (1+\dfrac{1}{k}) \\=\dfrac{1}{2}(1+0) \\=\dfrac{1}{2} \lt 1$
So, we can conclude that the given series converges by the ratio test.