Answer
Converges. The limit of the sequence seems to be $\pi$.
Work Step by Step
$a_n = 2tan^{-1}(1000n)$
$a_1 = 3.13959$
$a_2 = 3.14059$
$a_3 = 3.14093$
$a_4 = 3.14109$
$a_5 = 3.14119$
$a_6 = 3.14126$
$a_7 = 3.14131$
$a_8 = 3.14134$
$a_9 = 3.14137$
$a_{10} = 3.14139$
Converges; the sequence seems to get closer to $\pi$ as $n$ increases.