To see what's going on, replace the x=2014 with smaller numbers.
x=0
tan(arctan(1))=1
x=1
tan(arctan(1)+arctan(1/3))=2 [this was a recent problem solved via origami]
x=2
tan(arctan(1)+arctan(1/7))=3
So there's a pattern here. The answer would seem to be N=2015
The addition of a single term seems to make the answer go up by 1. Lets see what happens when we work backwards  try to make the answer go up by one to see what the term must be.
My calculator says tan(arctan(4)arctan(3))=1/13
which is promising. That's 3²+3+1
Lets try
tan(arctan(x+1)arctan(x)) = [(x+1)(x)]/[1+(x+1)(x)] =
(x^{2}+x+1)^{1}Bingo!
So this is the term that makes the answer increase by 1.
So I've proven N=x+1
So when x=2014, N=2015.

Posted by Jer
on 20140701 11:17:51 