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    I'm a little confused with this example of finding the limit to a convergent sequence:

    "A sequence is defined by the relation  a_{k+1} = -0.5 a_k +2 . Find L, the limit of  a_k as k tends to infinity."

    The solution they have given is:

     Here, f(a_k) = -0.5a_k +2

    Now take  L=f(L), giving  L = -0.5L +2

    This rearranges to  L =  \frac{4}{3}

    What I don't get is what they have done to work this out, and why that can get to the equation where L = 0.5L+2. Any explanation as on why they have done this would be appreciated- given the reccurance relation of any sequence, can I use the same meethod to find the limit if there is one. Also, how will I know if there is a limit, i.e is there a convergence or divergence test?
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    (Original post by Eux)
    I'm a little confused with this example of finding the limit to a convergent sequence:

    "A sequence is defined by the relation  a_{k+1} = -0.5 a_k +2 . Find L, the limit of  a_k as k tends to infinity."

    The solution they have given is:

     Here, f(a_k) = -0.5a_k +2

    Now take  L=f(L), giving  L = -0.5L +2

    This rearranges to  L =  \frac{4}{3}
    The point is that \displaystyle \lim_{k \rightarrow \infty} a_k = \lim_{k \rightarrow \infty} a_{k+1}.

    So *if* the limit exists, then it must have the same value, say L, in both cases.

    To show rigorously that it does indeed exist, you usually show (somehow) that the sequence a_k is increasing and bounded above, or something along those lines.
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    (Original post by atsruser)
    The point is that \displaystyle \lim_{k \rightarrow \infty} a_k = \lim_{k \rightarrow \infty} a_{k+1}.

    So *if* the limit exists, then it must have the same value, say L, in both cases.

    To show rigorously that it does indeed exist, you usually show (somehow) that the sequence a_k is increasing and bounded above, or something along those lines.
    Thank you. How would you tell if a sequence is convergent or divergent?
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    (Original post by Eux)
    Thank you. How would you tell if a sequence is convergent or divergent?
    If you only have a sequence defined by a recurrence relation, you have to use an argument similar to the one that I gave above. If you have a formula for the terms of the sequence (e.g. a_n= \frac{n}{n+1}) you have to examine its behaviour carefully (which can be tricky) using a variety of techniques.

    Are you an A level student or an undergraduate? If the former, then you won't have to worry about convergence issues much - you'll be told if the sequence converges or not, if necessary.
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    (Original post by atsruser)
    If you only have a sequence defined by a recurrence relation, you have to use an argument similar to the one that I gave above. If you have a formula for the terms of the sequence (e.g. a_n= \frac{n}{n+1}) you have to examine its behaviour carefully (which can be tricky) using a variety of techniques.

    Are you an A level student or an undergraduate? If the former, then you won't have to worry about convergence issues much - you'll be told if the sequence converges or not, if necessary.
    A level student revising for a Uni admissions test xD

    Is there a way to find the general formula of a sequence given by its recurrence relation?
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    Usually with these you show that the sequence is Cauchy ( e.g. contracting ) or that it is increasing / decreasing and bounded and hence must have a unique limit say L. Then solve for L. But I doubt this is useful for you...
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    (Original post by Eux)
    A level student revising for a Uni admissions test xD

    Is there a way to find the general formula of a sequence given by its recurrence relation?
    There are approaches that allow you to do this, but I don't recall the details too well, and I'm not sure if they work for all recurrence relations. Google is your friend here (generating functions is a good start).
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    (Original post by atsruser)
    There are approaches that allow you to do this, but I don't recall the details too well, and I'm not sure if they work for all recurrence relations. Google is your friend here (generating functions is a good start).
    Okay thanks for your help
 
 
 
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