Prove by method of differences that $\sum\limits_{r=1}^{n}{\frac{1}{\left( 2r-1 \right)\left( 2r+3 \

Question

Prove by method of differences that $\sum\limits_{r=1}^{n}{\frac{1}{\left( 2r-1 \right)\left( 2r+3 \right)}}=\frac{1}{3}-\frac{n+1}{\left( 2n+1 \right)\left( 2n+3 \right)}$ . [3] Hence, find the exact value of $\sum\limits_{r=5}^{\infty }{\frac{1}{\left( 2r-1 \right)\left( 2r+3 \right)}}$, [3] $\sum\limits_{r=0}^{n}{\frac{1}{\left( 2r+1 \right)\left( 2r+5 \right)}}$in terms of $n$. [2]

Accepted Answer

(ii)(a) $\dfrac{5}{99}$; (b) $\dfrac{1}{3} - \dfrac{n+2}{(2n+3)(2n+5)}$

2015 NJC P1 Q5

2015 NJC P1 Q5

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