NJC Sigma Notation Tutorial Q1
The method of differences can be used to evaluate $\sum\limits_{r=1}^{n}{\text{f}\left( r \right)}$, where
$\text{f}\left( r \right)=\frac{2}{r}-\frac{3}{r+1}+\frac{1}{r+2}$.
In each of the following situations, explain clearly the issue(s) that will arise if the series is evaluated as follows:
(a)
$\sum\limits_{r=1}^{n}{\text{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left( \frac{2}{r} \right)}-\sum\limits_{r=1}^{n}{\left( \frac{3}{r+1} \right)}+\sum\limits_{r=1}^{n}{\left( \frac{1}{r+2} \right)}=…$
(b)
$\sum\limits_{r=1}^{n}{\text{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left[ \frac{2\left( r+1 \right)\left( r+2 \right)-3r\left( r+2 \right)+r\left( r+1 \right)}{r\left( r+1 \right)\left( r+2 \right)} \right]}=…$
(c)
$\sum\limits_{r=1}^{n}{\text{f}\left( r \right)}=\sum\limits_{r=1}^{n}{\left( \frac{1}{r+2}+\frac{2}{r}-\frac{3}{r+1} \right)}=…$
Hence evaluate $\sum\limits_{r=1}^{n}{\text{f}\left( r \right)}$.
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