Series (Edexcel A Level Further Maths: Core Pure): Exam Questions

Prove that

$\sum _{r=1}^n\frac{1}{(r+1)(r+3)}=\frac{n(an+b)}{12(n+2)(n+3)}$

where $a$ and $b$ are constants to be found.

Use the method of differences to show that

$\sum _{r=1}^n\frac{1}{r(r+2)}=\frac{n(3n+5)}{4(n+1)(n+2)}$

Hence find the value of

$\sum _{r=1}^{\infty }\frac{1}{r(r+2)}$