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Math Help - Sequence question (analysis)

  1. #1
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    Sequence question (analysis)

    Hi guys,

    I'm working on a complex analysis problem which states that I have
    "a sequence {an} of distinct points in some region G such that an --> a in G as n --> infinity"

    (then I'll need to do something with the sequences)

    First question: Is {an} an infinite sequence? (In general, when I see this notation in analysis should I always assume an infinite sequence?)


    I'm going to assume that it is an infinite sequence, then, I have a function represented by, lets say, f(an) = cos(an)

    Since all sequences converge to the point a, is it true that f(an) = f(a) = cos(a)?
    Do I have a constant function? That's what it looks like to me.

    Thanks!


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  2. #2
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    Re: Sequence question (analysis)


    First question: Is {an} an infinite sequence? (In general, when I see this notation in analysis should I always assume an infinite sequence?)
    In this case, yes. In general, I do think of this as an infinite sequence. There may be exceptions, however, and depends on context.

    Since all sequences converge to the point a, is it true that f(an) = f(a)?
    Not in general. Consider the function f(x)=1 if x\not= a, f(x)=0 otherwise. The infinite sequence a_n=\frac{1}{n} has limit a=0 and satisfies f(a_n)=1 for all n. But f(0)=0. You need stronger conditions (i.e. function is continuous) for this statement to be true. For your example, it would be so since \cos(z) is analytic.
    Last edited by Gusbob; March 16th 2013 at 09:15 PM.
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  3. #3
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    Re: Sequence question (analysis)

    Thanks Gusbob.

    My function is, in fact, homomorphic on region G so I think that would make the previous statement true? Cos(zn) was just an example and it happened to be meet the criteria.

    EDIT: Saw your edit. Thanks.
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  4. #4
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    Re: Sequence question (analysis)

    One more question with regards to the original, convergent sequence {zn} of distinct points.
    Another problem asks whether there exists a function on region G such that f(zn) = n for all n.


    I have trouble interpreting what they're asking. What is n in the output? If input to function is a sequence then I would expect the output to be a sequence. Could n be the limit of the output sequence?

    (If the input to the function was a single element of the sequence, it would make more sense to me)

    Thanks
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