If You suppose that the sequence tends to a finite limit, the limit itself is obtained imposing that...

(1)

... i.e. the solution of the equation...

(2)

... so that is...

(3)

It is important to consider that the limit (3) isindependent fromif so that...

a) if then the sequence is monotonically increasing...

b) if then the sequence is constant...

c) if then the sequence is monotonically decreasing ...

Kind regards