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Homework Help: Three problems

  1. Mar 2, 2004 #1
    1. From a sequence An, collect even numbered terms En = A2n, and Odd terms O = A2n-1. SHow that An --> L iff En --> L and On --> L
    Im not sure if this is true while thinking about the proof but a sequence can behave in any way and therefore the even and odd terms may not necessairly come one after another so we can jus say for some n large enough En will approach the same thing as On. I'm clueless, please give me some idea

    2. well if that wasn't enough here's another one( |x| means absolute value)
    If An --> L then |An| --> L, is the converse true? That is If |An| -- > |L| then An --> L. Prove or give a counterexample. First of all is this even true? If it isn't then maybe a sequence like (-1/n)-1, maybe? Or does counterexample mean something else?

    3. If An --> L and Bn --> L then show that
    a1,b1,a2,b2,a3,b3,... converges to L. SO somehow we have to make sure that every term of B is greater than A somehow so obviously Bn> An but An+1 > Bn. I can think of a function that does this but how would you prove it?

    Any sort of guidance on ANY of these questions would be greatly appreciated
     
  2. jcsd
  3. Mar 3, 2004 #2

    HallsofIvy

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    Science Advisor

    "iff" works two ways. To show IF An--> L then En-->L and On-->L, you need to use the definition of limit: since An-->L, given &epsilon;>0, there exist N such that if n> N then |An-L|< &epsilon;
    Okay, now look at En and On. In order to get |En-L|< &epsilon; how large does n have to be? Same thing for On.

    To show IF En-->L and On-->L then An-->L, use the fact that "En-->L" means: for &epsilon;> 0, there exist N1 such that ....
    "On-->L" means: for &epsilon;> 0, there exist N2 such that.... Take N=larger of N1, N2 and use the fact that every member of An is in either En or On.

    What if L> 0? What if L< 0?!! The "theorem" stated is clearly incorrect. It should be "If An-->L then |An|--> |L|". Did you copy it correctly? What would the converse be?

    No, you DON'T need to do that. Nothing was said about this being an increasing sequence! Once again: An-->L means: given &epsilon;> 0, there exist N1 such that.... Bn-->L means: given &epsilon;> 0, there exist N2 such that....
    When you "fold" An and Bn together, you will need to look at N= larger of N1, N2.

    Do you see the similarity between this and #1?
     
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