The expression as given doesn't make much sense. I think you're trying to describe an infinitely nested radical. We can express this recursively by
Then you want to know the value of
if it exists.
To show the limit exists and that converges to some limit, we can try showing that the sequence is bounded and monotonic.
Boundedness: It's true that . Suppose
. Then
. So by induction,
is bounded above by 7 for all
.
Monontonicity: We have and
. It should be quite clear that
. Suppose
. Then
. So by induction,
is monotonically increasing.
Then because is bounded above and strictly increasing, the limit exists. Call it
. Now,
Solve for :
We omit because our analysis above showed that
must be positive.
So the value of the infinitely nested radical is 7.