5 Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { 2 ( 2 N + 3 ) }\).
Deduce that \(\sum _ { r = N + 1 } ^ { 2 N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } < \frac { 1 } { 8 N }\).
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5 Use the method of differences to show that $\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { 2 ( 2 N + 3 ) }$.
Deduce that $\sum _ { r = N + 1 } ^ { 2 N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } < \frac { 1 } { 8 N }$.
\hfill \mbox{\textit{CAIE FP1 2013 Q5}}