Question 9 - A-Level Maths

Exam Board	OCR
Module	FP1 (Further Pure Mathematics 1)
Year	2006
Session	January
Topic	Sequences and series, recurrence and convergence

Show that $\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }$.
Hence find an expression, in terms of $n$, for $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$
Hence find the value of
(a) $\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$,
(b) $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$.