1. (a) (4 points) determine whether the series ∑(k = 1 to ∞) k/(k² + 2) converges or diverges.

Explanation:

Step1: Use limit - comparison test

Compare $\sum_{k = 1}^{\infty}\frac{k}{k^{2}+2}$ with $\sum_{k = 1}^{\infty}\frac{1}{k}$. Calculate $\lim_{k\rightarrow\infty}\frac{\frac{k}{k^{2}+2}}{\frac{1}{k}}$.

Step2: Simplify the limit

$\lim_{k\rightarrow\infty}\frac{\frac{k}{k^{2}+2}}{\frac{1}{k}}=\lim_{k\rightarrow\infty}\frac{k\cdot k}{k^{2}+2}=\lim_{k\rightarrow\infty}\frac{k^{2}}{k^{2}+2}$. Divide both numerator and denominator by $k^{2}$: $\lim_{k\rightarrow\infty}\frac{1}{1 + \frac{2}{k^{2}}}=1$.

Step3: Recall p - series test

The series $\sum_{k = 1}^{\infty}\frac{1}{k}$ is a p - series with $p = 1$, and a p - series $\sum_{k=1}^{\infty}\frac{1}{k^{p}}$ diverges when $p\leq1$. Since $\lim_{k\rightarrow\infty}\frac{\frac{k}{k^{2}+2}}{\frac{1}{k}} = 1>0$ and $\sum_{k = 1}^{\infty}\frac{1}{k}$ diverges, by the limit - comparison test, $\sum_{k = 1}^{\infty}\frac{k}{k^{2}+2}$ diverges.

Answer:

The series $\sum_{k = 1}^{\infty}\frac{k}{k^{2}+2}$ diverges.