So recently I reviewed the concept of limit. I encountered a confusion: if a sequence has a common ratio of slightly over 1. It gradually approaches 1 as the ratio of the sequence is $$\sqrt{1/(b_n)^2+b_{n}^2}/(b_n)^2,$$
where $b_n$ is the previous term of the sequence. Thus the new number will be
$$\sqrt{1/(b_n)^2+b_{n}^2}/(b_n)$$
but it will also be greater than one. Suppose there are infinite terms in this sequence, will there be a limit of the sequence? What will it be?
Edit: the first term of the equence is 1. Only real number will be allowed in the sequence( both rational and rrational.) the exact sequence is $b_{n+1}=\sqrt{1/(b_n)^2+b_{n}^2}/(b_n)$ and there are infinte terms the sequence. Thank you for pointing out.