Answer
Converges
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)^{17}}{1/k^{17}}\\=\lim\limits_{k \to \infty} \dfrac{1}{(\dfrac{2k+3}{k})^{17}}\\=\lim\limits_{k \to \infty} \dfrac{1}{(2+\dfrac{3}{k})^{17}}\\=\dfrac{1}{2^{17}} \ne 0 \ne \infty$
So, we can conclude that the given series converges by the limit comparison test because $\Sigma_{n=1}^{\infty} \dfrac{1}{k^{17}}$ converges.
