Suppose a sequence is defined by , , and for . Prove converges and determine its limit.

So . This means that for all , there exists , such that whenever , then and .

Is this the right way to go about proving it?

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- Jul 5th 2008, 11:30 PMparticlejohnrecursive sequence
Suppose a sequence is defined by , , and for . Prove converges and determine its limit.

So . This means that for all , there exists , such that whenever , then and .

Is this the right way to go about proving it? - Jul 6th 2008, 12:21 AMIsomorphism
Yes I believe thats ok... Its induction isnt it?

Well there could be different ways to the limit, i tried the hardest way out there. I actually found out the sequence explicitly :p

There are different approaches, one of them is using generating functions and the other is by redefining the sequence and simple algebraic tricks.

I will do it using generating functions,

Let

This means

But for our series, . And .

Thus,

Clearly both can be represented as geometric progressions. And also the above function exists for some values of x(its called the region of convergence)

Thus:

By polynomial equality,

(Whew)

Now very clearly, . (Nerd)