I have one more analysis problem:
Let t_1 = 1 and t_n+1 = [1 - (1/(n+1)^2)]*t_n for n > 1.
a) Show that {t_n} converges.
b) Use induction to show that t_n = (n+1)/2n.
c) Find the limit of the sequence and prove that it is the limit.
I have one more analysis problem:
Let t_1 = 1 and t_n+1 = [1 - (1/(n+1)^2)]*t_n for n > 1.
a) Show that {t_n} converges.
b) Use induction to show that t_n = (n+1)/2n.
c) Find the limit of the sequence and prove that it is the limit.
I would start with part b), which you are told to prove by induction. The base case is easy to check, so what about the inductive step? We are told that , which you can simplify to (check that!).
If then , which simplifies to . That completes the inductive step.
Once you have done part b), you should be able to work out the limit of t_n as n→∞. That will deal with part a) and also help towards part c).