Let for and
I would like to prove that for the third derivative has a unique zero in . I thought calculating the third derivative and solving an equation, even if doable, doesn't seem very useful. I already "know" it's true after plotting many graphs, so all I would like to know is an elegant proof, not a proof that no sane person would like to read. (I have almost finished calculating the third derivative and if someone thinks it's a good idea, I can post the calculations. I can't guarantee they're correct though.)
The idea is to later take the sequence and show that if , then .
If somebody sees how to prove these in a way that requires a reasonable amount of paper, I would be very glad to see it. I've come up with the problem myself, so I don't know if there is a neat solution.
PS: I thought I would post the graph of a sample . Here it is for n=1000000+/-1 (I don't remember how I set the indices in the plotting program): View image: beztytu?u