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 {s_n} defined as s_1=\sqrt{2},\ s_{n+1}=\sqrt{2+\sqrt{s_n}}, show that {s_n} converges, and that it is bounded above by 2.


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  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 \sqrt{2+\sqrt{y}}=y (put y=s_{n+1}=s_n 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 #

  7. Now I see, somehow when I visit this page:
    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 #

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