## Nice Problem

Wednesday 17 November 2004 at 7:34 pm | In Articles | 9 Comments

Here’s a nice problem taken from ChapterZero who, in turn, got it from Rudin’s Principles of Mathematical Analysis

Given the sequence defined as , show that converges, and that it is bounded above by 2.

1. root(2+root(2+…=y and so root(2+y)=y so y^2-y-2=0 so y=2

Comment by scrambled — Friday 19 November 2004 2:58 am #

2. it’s actually from Principles aka Little Rudin.

Comment by 2718 — Friday 19 November 2004 7:23 am #

3. Thanks 2718. I have changed the link

Comment by Steve — Friday 19 November 2004 8:41 am #

4. scrambled: There are 2 problems. Firstly you have assumed that the sequence converges which you need to prove. Also the limit is given by (put in the recurrence relation) which gives approximately 1.83

Comment by Steve — Friday 19 November 2004 8:57 am #

5. hi steve,
why is it that when I visit your blog, the list of entries seemed to be different? But I can search for a specific entry and it exists.

meanwhile, thanks for making it easier to write maths in blog, i.e. the latexrender / mimetex info.

Comment by 2718 — Wednesday 1 December 2004 1:39 am #

6. I’m not sure I understand your question. The list of entries are different from what? I presume you are seeing things not in the way you expect – but in what way is it not right?

Comment by Steve — Wednesday 1 December 2004 8:10 am #

http://www.sixthform.info/maths/
I can see all the odd entries (counted in reverse chronological order), i.e I see the entries on 30th, 20th, 16th november, but not those on the 24th, 17th and 6th november. So when you write a new entry, all the even becomes odd and appears, while those previously odd entries disappear.
Cute. Is this deliberate?

Comment by 2718 — Wednesday 1 December 2004 9:13 am #

8. I have just viewed the site in Firefox and see the problem, which doesn’t show up in IE. Viewing the source (which I must say is very good in Firefox, with excellent syntax highlighting) showed a missing end of comment –>. IE didn’t care but Firefox does. I have fixed it now so I hope everything shows up properly (and for you?).
Thanks for letting me know!

Comment by Steve — Wednesday 1 December 2004 9:30 am #

9. yup, everything is fine now. cheers!

by the way, besides induction to show the sequence is monotone and bounded, are there other solutions to that Rudin problem?

Comment by 2718 — Thursday 2 December 2004 1:46 am #