Question 2 - A-Level Maths

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2017
Session	June
Topic	Sequences and series, recurrence and convergence

Verify that \(\frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 r + 1 ) ( 2 r + 3 ) } { ( r + 1 ) ( r + 2 ) } - \frac { ( 2 r - 1 ) ( 2 r + 1 ) } { r ( r + 1 ) } \right\}\).
Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 n + 1 ) ( 2 n + 3 ) } { ( n + 1 ) ( n + 2 ) } - \frac { 3 } { 2 } \right\}\).
Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) }\).