This is an interesting sequence!

The answer to 1) has to be No: it is not possible (as far as I can see) to give an explicit formula for You just have to work with the recursive formula that gives you in terms of

For 2) also, it does not seem possible to find the limit explicitly. The best that can be done is show that the limit must exist, and that it can be narrowed down to lie in some interval. The value of the limit will depend on the first term of the sequence.

From the equation , you can show by induction that and also that Thus is a decreasing sequence of positive terms. There is a theorem which says that any decreasing sequence that is bounded below must converge to a limit. So we know that this sequence must converge to some limit The big question is whether this limit is actually equal to 0, or whether the sequence converges to a strictly positive limit. In fact, it is true that

To see that, write for the first term of the sequence, so that Since the sequence decreases, it follows that for all n. Therefore for each n=1,2,3,... . It follows that

(sum of geometric series). If you now let on both sides, you find that Therefore

Thus the limit of the sequence is strictly positive, and lies between and I don't see any way of narrowing it down more closely than that.