$1 + \sum\limits_{r = 0}^{22} {\left\{ {r\left( {r + 2} \right) + 1} \right\}} \cdot r! = k!$,then the number of divisors of $k$ is

The value of the infinite product $\prod\limits_{n = 2}^\infty {\left( {1 - \frac{1}{{{n^2}}}} \right)}$ is

The value of $\sum\limits_{k = 1}^\infty {\frac{{3{k^2} + 3k + 1}}{{{{\left( {{k^2} + k} \right)}^3}}}} $ is equal to

If $\sum_{r=1}^{25} \left( \frac{r}{r^{4}+r^{2}+1} \right) = \frac{p}{q}$ where $p$ and $q$ are positive integers such that $\gcd(p,q)=1$,then $p+q$ is equal to . . . . . . .

$\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to

$\text{Given, } \frac{\sin 1^{\circ}}{\sin x^{\circ} \sin (x+1)^{\circ}} = \cot x^{\circ} - \cot (x+1)^{\circ}, \text{ then the value of } \frac{1}{\sin 45^{\circ} \sin 46^{\circ}} + \frac{1}{\sin 46^{\circ} \sin 47^{\circ}} + \dots + \frac{1}{\sin 89^{\circ} \sin 90^{\circ}} \text{ is}$