| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2013 |
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| Session | June |
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| Topic | Sequences and series, recurrence and convergence |
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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 \(\sum _ { r = N + 1 } ^ { 2 N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } < \frac { 1 } { 8 N }\).