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?
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
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,
Now very clearly, .