Determine whether the series $\sum\limits_{k\,=\,1}^{\infty }a_{k}=$ $\sum\limits_{k=1}^{\infty }\dfrac{2k}{k^{2}+1}$ converges or diverges.

Solution The function $f(x)=\dfrac{2x}{x^{2}+1}$ is continuous, positive, and decreasing since $f' (x) \le 0$ for all numbers $x\,{\geq}\,1$, and $a_{k}\,{=}\,f(k)$ for all positive integers $k.$ Using the Integral Test, we find $$\begin{eqnarray*} \int_{1}^{\infty }\frac{2x}{x^{2}+1}\,dx\! : \lim_{b\,\rightarrow \,\infty }\int_{1}^{b}\frac{2x}{x^{2}+1}\,dx&=&\lim_{b\,\rightarrow \,\infty }\,\left[\ln (x^{2}+1)\right] _{1}^{b} \\[8pt] &=& \lim_{b\,\rightarrow \,\infty }\,[\ln(b^{2}+1)-\ln 2] =\infty \end{eqnarray*}$$

Since the improper integral diverges, the series $\sum\limits_{k=1}^{\infty}\dfrac{2k}{k^{2}+1}$ also diverges.