The sum $1(1!) + 2(2!) + 3(3!) + \dots + n(n!)$ equals

Assertion $(A)$: $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots+(81+90+100)=1000$
Reason $(R)$: $\sum_{r=1}^n(r^3-(r-1)^3)=n^3$ for any natural number $n$.

The sum of the series $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots$ up to $15$ terms is

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

$\lim _{n \rightarrow \infty} \sum_{r=1}^n \tan ^{-1}\left(\frac{2 r}{r^4+r^2+2}\right) = $

$\frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \frac{1}{7 \cdot 9} + \ldots$ to $24$ terms $=$