Answer
Inconclusive
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^2+1)}{k^2+2k+2} \times \dfrac{k^2+1}{k} \\=\lim\limits_{k \to \infty} \dfrac{(k+1)}{k^2+2k+2} \times \lim\limits_{k \to \infty} \dfrac{k^2+1}{k} \\=\lim\limits_{k \to \infty} \dfrac{1+1/k^2}{1+2/k+2/k^2} \times \lim\limits_{k \to \infty} (1+\dfrac{1}{k}) \\=\dfrac{1+0}{1+0+0} \times (1+0) \\=1$
So, we can conclude that the given series is inconclusive by the ratio test.
