I’m sorry, but I have to disagree with all the previous answers except that of @JanEerland. It seems to me that the infinite expression that you have written can be interpreted in only one way, as the limit of a sequence, which we must show to be convergent. If we do this, the limit is unique.
The sequence is defined recursively as follows:
One sees easily that $a_n<2$ for all $n$, and a little less easily that the sequence is increasing. Your computation gives two possible values, but only one of these is $\le2$, and hence that one is the value, to the extent that the expression is to be viewed as a limit.