Answer
The sequence converges to $0$.
Work Step by Step
$a_n = cot^{-1}(2^n)$
$a_1 = 0.463638$
$a_2 = 0.244979$
$a_3 = 0.124355$
$a_4 = 0.0624188$
$a_5 = 0.0312398$
$a_6 = 0.0156237$
$a_7 = 0.00781234$
$a_8 = 0.00390623$
$a_9 = 0.00195312$
$a_{10} = 0.000976562$
The sequence seems to approach $0$ as $n$ increases.