Using the Integral Test

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.