anonymous
  • anonymous
Find the limit of the following sequence (I'll post it below)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, ...\] So in recursive form it would look like this: \[a_n=\sqrt{2(a_n-1)}\] I know that the answer to this is supposed to be 2, but I'm not sure how to get there. Any help would be appreciated.
anonymous
  • anonymous
put \(x=\sqrt{2\sqrt{2\sqrt2}..}\)
anonymous
  • anonymous
then square it

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anonymous
  • anonymous
you get \(x^2=2x\)
experimentX
  • experimentX
my intuition says that you need to show that the limit exists prior to calculating it.
anonymous
  • anonymous
@satellite73 Thanks so much, I don't know why I didn't think of that. We're just starting out with sequences and series and for some reason I find them a little tricky to work with. @experimentX I was told that the limit existed so I didn't really need to determine that, but for next time it would probably help to know how to do that. Do you know how to show that this limit exists?
anonymous
  • anonymous
@satellite73 Doesn't x=0 also fulfill that equation? How do I know the limit isn't 0?
experimentX
  • experimentX
show the sequence is monotonically increasing and is bounded above.

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