Limit/summation problem, help appreciated
I spent the night trying to test a theory of mine (irrelevant), which brought me to this limit:
EDIT: Here's a picture, much easier:
I expect the answer to be π-0.5
(≈ 0.564), but my approximations up to 10000 iterations are closer to 0.574, off by an entire hundredth. I need a way to compute the exact limit as N approaches infinity, but I can't seem to simplify the summation any further than I already have (which is an improvement on the original).
I know that plenty of you must have taken some calculus - maybe you know the next step here. Any ideas?
Note: Since this is not a problem from any textbook, it may very well be the case that this cannot
be simplified. But, I simply refuse to believe that right now.