Official solution
\[{1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..........\infty \]
\[\left| {{T_n}} \right| = {1 \over {n\left( {n + 1} \right)}} = \left( {{1 \over n} - {1 \over {n + 1}}} \right)\]
\[S = {T_1} - {T_2} + {T_3} - {T_4} + {T_5}.........\infty \]
\[ = \left( {{1 \over 1} - {1 \over 2}} \right) - \left( {{1 \over 2} - {1 \over 3}} \right) + \left( {{1 \over 3} - {1 \over 4}} \right)\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {{1 \over 4} - {1 \over 5}} \right).......\]
\[ = 1 - 2\left[ {{1 \over 2} - {1 \over 3} + {1 \over 4} - {1 \over 5}.........\infty } \right]\]
\[ = 1 - 2\left[ { - \log \left( {1 + 1} \right) + 1} \right]\]
\[ = 2\log 2 - 1 = \log \left( {{4 \over e}} \right)\]