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MarcLeclair

  • 2 years ago

Can anybody help me with this sequence question?

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  1. MarcLeclair
    • 2 years ago
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    I tried to use proof by induction by I don't know how to show its bounded above x.x I used the rule \[a _{1}= \sqrt{2} , a _{n+1}= \sqrt{2a _{n}}\]

  2. electrokid
    • 2 years ago
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    did you try the ratio test?

  3. electrokid
    • 2 years ago
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    \[L=\lim_{n\to\infty}\left|a_{n+1}\over a_n\right|=\lim_{n\to\infty}\sqrt{2}>1\]

  4. MarcLeclair
    • 2 years ago
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    Oops my post erased. My book / cal class does not and will not go over the ratio test I believe. I found the equation myself ( i mean the formula), when I use \[\lim_{n \rightarrow \infty} f(a _{n} ) = f(l)\], the answer it gives me is right but induction doesn't seem to work because I don't know if its bounded above ( well I'm assuming its above because it's increasing)

  5. Zarkon
    • 2 years ago
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    use induction to show that \[a_n\le 2\] for all \(n\)

  6. MarcLeclair
    • 2 years ago
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    So I just calculate its lowest upper bound limit by plugging in n?

  7. MarcLeclair
    • 2 years ago
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    oh nvm it cant be greater than 2 because 2= root of 4, right?

  8. Zarkon
    • 2 years ago
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    \[\sqrt{2}\le 2\] assume \(a_n\le 2\) \(a_{n+1}=\sqrt{2\cdot a_n}\le \sqrt{2\cdot 2}=2\)

  9. Zarkon
    • 2 years ago
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    then use monotone convergence theorem

  10. MarcLeclair
    • 2 years ago
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    makes very much sense. Thanks a lot, I forgot to think of that. ^^ and yeah, a function is bounded above if its increasing.

  11. Zarkon
    • 2 years ago
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    "...a function is bounded above if its increasing" that is not true

  12. MarcLeclair
    • 2 years ago
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    I mean sequence* I thought if it was a sequence that was only increasing it was bounded above

  13. Zarkon
    • 2 years ago
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    \[a_n=n\] is an increasing sequence

  14. Zarkon
    • 2 years ago
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    it is not bounded above

  15. MarcLeclair
    • 2 years ago
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    So, is it IF a sequence is bounded, it is converging? Im sorry Im pretty new to this stuff

  16. Zarkon
    • 2 years ago
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    if the sequence is bounded and it is either increasing or decreasing then the sequence converges.

  17. MarcLeclair
    • 2 years ago
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    thank you for your patience!

  18. Zarkon
    • 2 years ago
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    Monotone convergence theorem

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