Question 5 - A-Level Maths

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

Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
Hence find an expression in terms of $n$ for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
State the value of
(a) $\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }$,
(b) $\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }$.