Question

does the series from n=1 to infinity of 1/arctan(2n) converge?

I said yes but I didn't know what to

Answer 1

Assuming that your arctan numbers are between 0 and 2 pi, regardless of what (2n) is, you will end up with an infinite number of numbers each of which approaches 1/arctan(infinity) = (pi/2)

The sum of that series will certainly diverge.