$\sum_{n=1}^{10} \left( \frac{528}{n(n+1)(n+2)} \right)$ is equal to:

If $t_{n} = \frac{1}{4}(n+2)(n+3)$ for $n = 1, 2, 3, \dots$,then find the value of $\frac{1}{t_{1}} + \frac{1}{t_{2}} + \frac{1}{t_{3}} + \dots + \frac{1}{t_{2003}}$.

$\sum_{k=0}^{12} \frac{1}{\sin \left((k+1) \frac{\pi}{6}+\frac{\pi}{4}\right) \sin \left(\frac{k \pi}{6}+\frac{\pi}{4}\right)} = $

If $\sum_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}$,where $m$ and $n$ are coprime,then $m+n$ is equal to.

Find the sum of the $n$ terms of the series $\frac{3}{1^2} + \frac{5}{1^2 + 2^2} + \frac{7}{1^2 + 2^2 + 3^2} + ...$

The sum of the infinite series ${\tan ^{ - 1}}\left( {\frac{2}{{1 - {1^2} + {1^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{4}{{1 - {2^2} + {2^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{6}{{1 - {3^2} + {3^4}}}} \right) + \dots$ is