Limit of a sequence of fractionsI have that: S n = b ⋅ S n...
Limit of a sequence of fractions
I have that:
So, for we will get:
And for :
And so on. Now my question is: what happens when ?
Answer & Explanation
Assume that is associated with . The recurrence
can be written in the following form
hence the closed form of both the and the sequences just depends on the diagonalization of , whose eigenvalues are given by .
for some constants depending on the initial values. By taking the limit as , we simply get . On the other hand, by assuming as , such limit has to fulfill
hence the only chances are