I'm a little confused with this example of finding the limit to a convergent sequence:
"A sequence is defined by the relation. Find L, the limit of
as k tends to infinity."
The solution they have given is:
Now take, giving
This rearranges to
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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Eux
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- 20-10-2015 18:21
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- 21-10-2015 17:32
(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. Find L, the limit of
as k tends to infinity."
The solution they have given is:
Now take, giving
This rearranges to.
So *if* the limit exists, then it must have the same value, say, in both cases.
To show rigorously that it does indeed exist, you usually show (somehow) that the sequenceis increasing and bounded above, or something along those lines.
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Eux
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- 21-10-2015 17:50
(Original post by atsruser)
The point is that.
So *if* the limit exists, then it must have the same value, say, in both cases.
To show rigorously that it does indeed exist, you usually show (somehow) that the sequenceis increasing and bounded above, or something along those lines.
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- 21-10-2015 18:01
(Original post by Eux)
Thank you. How would you tell if a sequence is convergent or divergent?) 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. -
Eux
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- 21-10-2015 18:05
(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.) 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.
Is there a way to find the general formula of a sequence given by its recurrence relation? -
hassassin04
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- 21-10-2015 18:10
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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- 21-10-2015 18:18
(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? -
Eux
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- 21-10-2015 19:09
(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).
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