Monotone and Cauchy Sequences

[bearej50]

Let s1 = ?(6), s2 = ?(6+?(6)), s3 = ?(6+?(6+?(6))), and in general define sn+1 = ?(6+sn). Prove that (sn) converges, and find its limit.


*note: "?" means "the square root of"

 

[DrMike]

Your title gives a hint - can you prove that s[sub:31k2w1lm]n+1[/sub:31k2w1lm] > s[sub:31k2w1lm]n[/sub:31k2w1lm] for all n ? Then you've proved it's monotonically increasing...

Then can you find an upper bound for it? Find some number U that lets you prove that if s[sub:31k2w1lm]n[/sub:31k2w1lm] < U, then s[sub:31k2w1lm]n+1[/sub:31k2w1lm] < U also (and note that s[sub:31k2w1lm]1[/sub:31k2w1lm]<U). Then you've proved it's bounded above.

A sequence that's monotonically increasing and bounded above must converge. It has a limit.

Then, if the limit is L, what happens when you take the limit of both sides of \(\displaystyle s_{n+1}=\sqrt{6+s_n}\) ? You should get an equation for L, which you can solve...

 

[bearej50]

I follow your reasoning. I am still having trouble properly proving this though. I believe that I would use math induction...