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 $\displaystyle t_{n+1} = t_n\Bigl(1-\frac1{(n+1)^2}\Bigr)$, which you can simplify to $\displaystyle t_{n+1} = t_n\frac{n(n+2)}{(n+1)^2}$ (check that!).
If $\displaystyle t_n = \frac{n+1}{2n}$ then $\displaystyle t_{n+1} = \frac{(n+1)n(n+2)}{2n(n+1)^2}$, which simplifies to $\displaystyle \frac{n+2}{2(n+1)}$. 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).