Is this number computable?

Yes, this number is computable. Your definition of it is an algorithm for computing its digits.

More generally, you should be aware that not knowing what a number's value is has little to do with whether the number is computable. For instance, define a number $n$ as follows. If ZFC is consistent, $n=1$. If ZFC is inconsistent, $n=0$. This number is certainly computable: either the program that just outputs $1$ computes it, or the program that just outputs $0$ computes it. It doesn't matter that we can't determine (in ZFC) which of these programs is the right program to use: either way, there exists a program that works.

Of course, this value is either $0$ or $2^{-n}$ for some $n$, and all of those numbers are computable, so this number is computable. (Assuming the result is treated as binary.)

If base $10$, then this value is either $0$ or $9^{-1}\cdot 10^{-n}$ for some $n$, and any rational number is computable.