Worked Example 4

(i)

By considering \(\text{f}\left( r \right)-\text{f}\left( r+1 \right)\), where \(\text{f}\left( r \right)=\frac{1}{r}\), find the sum to \(2n\) terms of the series \(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+…\)

(ii)

Show that the sum to \(2n\) terms of the series \(\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}}+\frac{1}{{{4}^{2}}}+\frac{1}{{{5}^{2}}}+…\) is less than \(1\).