Answer
Diverges
Work Step by Step
Apply the limit comparison test.
Therefore, $ \lim\limits_{k \to \infty} \dfrac{a_k}{b_k}=\lim\limits_{k \to \infty} \dfrac{1/2k+3}{1/k}\\=\lim\limits_{k \to \infty} \dfrac{k}{2k+1}\\=\lim\limits_{k \to \infty} \dfrac{1}{2+3/k}\\=\dfrac{1}{2+0}\\=\dfrac{1}{2} \ne 0 \ne \infty$
So, we can conclude that the given series diverges by the limit comparison test because $\Sigma_{n=1}^{\infty} \dfrac{1}{k}$ is a p-series with $p=1$, so it diverges.
