calculus help?

Question

A sequence {an} is given by a1= (radical 2) and a(n+1)= radical (2 + an).

a) by induction or otherwise, show that {an} is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that lim n -> infinity of {an} exists.

b) Find lim n -> infinity of {an}

Answer 1

Okay, Doctor , I'll give you a hint but you have to start doing your own work. Look at the ratio of a(n+2) to a(n+1). I think you can easily show that a(n+2)/a(n+1) > 1.

A proof that a(n) < 3 by induction simply involves observing that sqrt(2 + 3) < 3 .

The Monotonic Sequence Theorem says that if a sequence is increasing and bounded above, it has a limit as n-> inf. When you complete the above, you will have the hypothesis satisfied.