NJC Sigma Notation Tutorial Q1

The method of differences can be used to evaluate \(\sum\limits_{r=1}^{n}{\mathrm{f}\left( r \right)}\), where

\(\mathrm{f}\left( r \right)=\frac{2}{r}-\frac{3}{r+1}+\frac{1}{r+2}\).

In each of the following situations, explain clearly the issue(s) that will arise if the series is evaluated as follows:

1. \(\sum\limits_{r=1}^{n}{\mathrm{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left( \frac{2}{r} \right)}-\sum\limits_{r=1}^{n}{\left( \frac{3}{r+1} \right)}+\sum\limits_{r=1}^{n}{\left( \frac{1}{r+2} \right)}=...\)
2. \(\sum\limits_{r=1}^{n}{\mathrm{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left[ \frac{2\left( r+1 \right)\left( r+2 \right)-3r\left( r+2 \right)+r\left( r+1 \right)}{r\left( r+1 \right)\left( r+2 \right)} \right]}=...\)
3. \(\sum\limits_{r=1}^{n}{\mathrm{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left( \frac{1}{r+2}+\frac{2}{r}-\frac{3}{r+1} \right)}=...\)

Hence evaluate \(\sum\limits_{r=1}^{n}{\mathrm{f}\left( r \right)}\).