Use the method of differences to show that \(\displaystyle\sum_{r=2}^{n}\ln\left[\frac{(r-1)(r+1)}{r^2}\right]=\ln\left(\frac{n+1}{n}\right)-\ln 2\). Show that the corresponding infinite series is convergent and state the sum to infinity. Show that \(\displaystyle\sum_{r=10}^{20}\ln\left[\frac{(r-1)(r+1)}{r^2}\right]=\ln\left(\frac{a}{b}\right)\), where \(a\) and \(b\) are integers to be found.

Use the method of differences to show that \(\displaystyle\sum_{r=2}^{n}\ln\left[\frac{(r-1)(r+1)}{r^2}\right]=\ln\left(\frac{n+1}{n}\right)-\ln 2\).

Show that the corresponding infinite series is convergent and state the sum to infinity.

Show that \(\displaystyle\sum_{r=10}^{20}\ln\left[\frac{(r-1)(r+1)}{r^2}\right]=\ln\left(\frac{a}{b}\right)\), where \(a\) and \(b\) are integers to be found.

N2023 P1 Q5

N2023 P1 Q5