In Barry Cooper's "Computability theory" there is an exercise (16.4.12 to be precise) which I can't find anyway to do it... it states:
Let $A \subseteq \mathbb{N}$. Define
$$r_A=\sum_{i \in A} 2^{-i}$$
Prove that $r_A$ is computable if, and only if, $A$ is computable. I'm having trouble understanding the notion of modulus of convergence, so I don't know how to prove the statement "if $A$ is computable, then $r_A$ is computable". Do I have to use Church-Turing thesis?
I appreciate any help!