2019 NYJC CT2 P1 Q3

Using integration by parts, find \(\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{n}{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}\) in terms of \(n\) and \(\mathrm{e}\).[3]

The diagram below shows a part of the curve with equation \(y=\frac{\ln x}{{{x}^{5}}}\). An infinite sequence of rectangles are formed using points on the curve as shown in the diagram.

Show that the total area of the rectangles is given by \(\sum\limits_{k=3}^{\infty }{\frac{\ln k}{{{k}^{5}}}}\).[1]

The Monotone Convergence Theorem for a series states that if \({{a}_{k}}>0\) for all \(k\) and \(\sum{{{a}_{k}}\le M}\) for some \(M\in \mathbb{R}\), then \(\sum{{{a}_{k}}}\) is convergent. By considering the integral \(\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{\infty }{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}\) and the above graph, show that the series \(\sum\limits_{k=3}^{\infty }{\frac{\ln k}{{{k}^{5}}}}\) is convergent.[4]