2020 NJC P1 Q6

The integral test for convergence, known as the Maclaurin-Cauchy test, is a method used to test for convergence of an infinite series of non-negative terms. The Maclaurin-Cauchy test states that

‘For a non-negative and non-increasing function \(\mathrm{f}\) defined on the interval \([1,\infty )\), a series of the form \(\sum\limits_{r=1}^{\infty }{\mathrm{f}(r)}\) converges if \(\displaystyle\displaystyle \displaystyle\int_{1}^{\infty }{\mathrm{f}(x)\,\mathrm{d}x}\) is finite.

1. Show that \(\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}\) is positive and decreasing on the interval \([1,\infty )\).[3]
2. Use the Maclaurin-Cauchy test to determine if \(\sum\limits_{r=1}^{\infty }{\frac{{{\tan }^{-1}}r}{1+{{r}^{2}}}}\) converges.[5]