Using the Integral Test

Determine whether the series \(\sum\limits_{k\,=2}^{\infty }\dfrac{1}{k(\ln k) ^{2}}\) converges or diverges.

Solution The function \(f(x)\,{=}\,\dfrac{1}{x (\ln x)^{2}}\) is continuous, positive, and decreasing on the interval \([2,\infty)\). Also \(\dfrac{1}{k(\ln k) ^{2}}\,{=}\,f(k) \) for all integers greater than or equal to \(2.\) Using the Integral Test, we investigate the improper integral \(\int_{2}^{\infty }\dfrac{dx}{x(\ln x)^{2} }.\) We find an antiderivative of \(\dfrac{1}{x(\ln x) ^{2}}\) by using the substitution \(u=\ln x,\) \(du=\dfrac{1}{x}dx.\) Then, \[ \int \frac{dx}{x(\ln x)^{2}}=\int \frac{du}{u^{2}}=-\frac{1}{u}+C=-\frac{1 }{\ln x}+C \]

Now we find the improper integral \[ \int_{2}^{\infty }\frac{dx}{x(\ln x)^{2}}\!:\,\,\lim_{b\,\rightarrow \,\infty }\,\left[ -\frac{1}{\ln x} \right] _{2}^{b}=\lim_{b\, \rightarrow \,\infty }\,\left( -\frac{1}{\ln b}\right) +\frac{1}{\ln 2}=\frac{1 }{\ln 2} \]

Since the improper integral \(\int_{2}^{\infty }\dfrac{dx}{x(\ln x)^{2}}\) converges, the series \(\sum\limits_{k\,=2}^{\infty }\dfrac{1}{k(\ln k) ^{2}}\) converges.