Using partial fractions, show that \(\sum\limits_{n=1}^{N}{\frac{1}{n\left( n+1 \right)\left( n

Question

Using partial fractions, show that $\sum\limits_{n=1}^{N}{\frac{1}{n\left( n+1 \right)\left( n+2 \right)}=\frac{1}{4}-\frac{1}{2\left( N+1 \right)\left( N+2 \right)}}$.[4] Deduce the value of $\sum\limits_{n=2}^{\infty }{\frac{1}{n\left( n+1 \right)\left( n+2 \right)}}$.[2] Find $\sum\limits_{n=1}^{N-1}{\frac{1}{\left( n+1 \right)\left( n+2 \right)\left( n+3 \right)}}$ in terms of $N$.[2]

Accepted Answer

(i) Shown; (ii) 1/12; (iii) 1/12 − 1/(2(N+1)(N+2))

2010 NYJC Promo Q5 [Modified]

2010 NYJC Promo Q5 [Modified]

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