\( \sum_{i=1}^{n+1}\frac{1}{(2i-1)(2i+1)}=\sum_{i=1}^{n}\frac{1}{(2i-1)(2i+1)}+\frac{1}{(2n+1)(2n+3)}=\frac{n}{2n+1}+ \frac{1}{(2n+1)(2n+3)} \)
\(= \frac{n(2n+3)+1}{(2n+1)(2n+3)}=\frac{2n^2+3n+1}{(2n+1)(2n+3)}=\frac{(n+1)(2n+1)}{(2n+1)(2n+3)}=\frac{n+1}{2n+3}=\frac{n+1}{2(n+1)+1} \)
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