By definition, a continued fraction has a value if and only if the sequence of the so-called convergents (the results of the iterations) converges.

The fact that you can get an equation that must be fulfilled in case that the continued fraction has a value does not mean that we can assign a value to that continued fraction. In other words:

"continued fraction has value $\Longrightarrow$ equation holds for that value"

but not the other way round.

Your sequence of convergents does not converge. Therefore there is no value associated with the continued fraction.

It is similar with other sequences or series. Famous example, you can perform manipulations on the series $\sum_{n=1}^{\infty} n$ which make it seem that its value is $-\frac{1}{12}$, but this is obviously nonsense when we apply the usual meaning of summation.

By definition, a continued fraction has a finite value if and only if the sequence of the so-called convergents (the results of the iterations) converges.

The fact that you can get an equation that must be fulfilled in case that the continued fraction has a finite value does not mean that we can assign a finite value to that continued fraction. In other words:

"continued fraction has finite value $\Longrightarrow$ equation holds for that value"

but not the other way round.

Your sequence of convergents does not converge. Therefore there is no finite value associated with the continued fraction.

It is similar with other sequences or series. Famous example, you can perform manipulations on the series $\sum_{n=1}^{\infty} n$ which make it seem that its value is $-\frac{1}{12}$, but this is obviously nonsense when we apply the usual meaning of summation.