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# Finding the limit of a sequence? watch

1. 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?
2. (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
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 sequence is increasing and bounded above, or something along those lines.
3. (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 sequence is increasing and bounded above, or something along those lines.
Thank you. How would you tell if a sequence is convergent or divergent?
4. (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. ) 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.
5. (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.
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?
6. 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...
7. (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).
8. (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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