I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$
On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan form as $$S= 23\tan^{-1}(1/3)+22\tan^{-1}(1/5) + \cdots + \tan^{-1}(1/47)$$ How to sum this series?