I have to determine whether the sequence converges or diverges and, if it does converge, find the limit. I'm confused about these two.
The limit of arctan(x) as x goes to infinity is pi/2. So, doesn't arctan(2x) do the same, just the behavior around the origin is altered? Can I just put that the limit of arctan(2x) is pi/2 even if I really didn't do anything to prove it?
Both go to infinity, the denominator goes just a bit quicker. So, I do l'hopital rule and I get
1/x * 2x/2
Which reduces to 1 if I'm not mistaken. So, I get the obvious limit.
lim 1 = 1
which means my original problem's limit is 1. Is that right? It just seemed to easy to me.
1. Firstly let . Then and . So then . Furthermore note that . So we have show that is both bounded and monotonic. Thus it is convergent and its limit is
2. Next let
So now consider that and that . So then . And notice that . Therefore we have shown that the sequence is bounded and monotonic, thus convergent. And then we state that .