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{1/9k+6}{1/k}\\=\lim\limits_{k \to \infty} \dfrac{1}{9+\dfrac{6}{k}}\\=\dfrac{1}{9+0}\\=\dfrac{1}{9} \ne 0 \ne \infty$
So, we can conclude that the given series diverges by the limit comparison test because $\Sigma_{n=1}^{\infty} \dfrac{1}{k}$ diverges.
