2017 NYJC Promo Q5
(i)
By considering $\tan \left( A-B \right)$ or otherwise, show that
${{\tan }^{-1}}\left( k+1 \right)-{{\tan }^{-1}}\left( k \right)={{\cot }^{-1}}\left( {{k}^{2}}+k+1 \right)$ for $k>0$.
[2]
(ii)
Hence find $\sum\limits_{k=1}^{n}{{{\cot }^{-1}}\left( {{k}^{2}}+k+1 \right)}$ in terms of $n$.
[3]
(iii)
Deduce the exact value of $\sum\limits_{k=1}^{\infty }{{{\cot }^{-1}}\left( {{k}^{2}}+k+1 \right)}$.
[2]
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