Does a terminating recurrence relation diverge? Given the recurrence relation
$$u_1=-3.25 \ \& \ u_{k+1}=\frac{4}{u_k+2}$$
is $\{u_k\}$ convergent?
A quick check for the definition of convergence gives the following:
If $\forall \epsilon \ \exists L\exists N$ s.t. $\forall n>N$, $|u_n-L|<\epsilon$, then $\{u_k\}$ is convergent. Otherwise, it is divergent.
Since no such $(L,N)$ exists, it is divergent.
However, by listing out the first 6 terms, $\{-3.25,-3.2,-3\frac{1}{3},-3,-4,-2\}$, we realise that $u_k$ is undefined when $k\geq7$.
Does it even make sense to discuss convergence of a terminating recurrence relation?

If you speak about a recurrence relation being convergent, that is actually just shorthand for speaking about whether the sequence defined by the recurrence relation converges or not.
In order to converge, a sequence needs to be infinite. It is allowed for its terms to be all identical from a certain point on, but there does needs to be an infinity of terms in the first place.
In this case, your recurrence does not define an infinite sequence at all. Therefore, in particular, it doesn't define any converging sequence, so it would be be wrong to say that the recurrence "converges".
