Use the method of differences to find $\sum\limits_{r=1}^{n}{\ln \left( \frac{r}{r+2} \right)}$ in

Question

Use the method of differences to find $\sum\limits_{r=1}^{n}{\ln \left( \frac{r}{r+2} \right)}$ in the form $\ln \left[ \frac{k}{\left( n+1 \right)\left( n+2 \right)} \right]$, where $k$ is a constant to be determined. Hence find $\ln \left( \frac{n+1}{n+3} \right)+\ln \left( \frac{n+2}{n+4} \right)+...+\ln \left( \frac{3n}{3n+2} \right)$.

Accepted Answer

$k=2$, $\ln \frac{(n+1)(n+2)}{(3n+1)(3n+2)}$

Method of Difference Q4

Method of Difference Q4

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