Given that \(\sum\limits_{r=0}^{N}{\frac{1}{\left( r+1 \right)\left( r+3 \right)}}=\frac{3}{4}-

Question

Given that $\sum\limits_{r=0}^{N}{\frac{1}{\left( r+1 \right)\left( r+3 \right)}}=\frac{3}{4}-\frac{1}{2\left( N+2 \right)}-\frac{1}{2\left( N+3 \right)}$, find $\sum\limits_{r=4}^{N+3}{\frac{1}{\left( r-1 \right)\left( r-3 \right)}}$ $\sum\limits_{r=5}^{N}{\frac{1}{\left( r-1 \right)\left( r-3 \right)}}$

Accepted Answer

(i) $\frac{3}{4}-\frac{1}{2(N+1)}-\frac{1}{2(N+2)}$ (ii) $\frac{5}{12}-\frac{1}{2(N-2)}-\frac{1}{2(N-1)}$

Sigma Notation By Substitution Q1

Sigma Notation By Substitution Q1

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