If you can find arbitrarily large values of n such that then clearly the sequence will not tend to infinity. So we need to look at when

To get sufficiently small, we need n to be within approximately 1/n of an odd multiple of Thus the question becomes: can we find arbitrarily large values of n for there exists an integer k such that or equivalently ? That is a question about rational approximations of , and problems like that are not easy. You can find a bit more about this in my comments #5 and #10 in this thread.