I think -- and hope -- that every computer science student is confronted with this problem which feels like a paradoxon. It is a very good example for the difference of computable in TCS sense and computable in a practical sense.
My thoughts back then were: "Yea, if I knew the answer, it would obviously be computable. But how to find out?" The trick is to rid yourself from the illusion that you have to find out wether $\pi$ has this property or not. Because this, obviously (read: imho), cannot be done by a Turing machine (as long as we do not have more knowledge than we have about $\pi$).
Consider your definition for computability: we say $f$ is (Turing-)computable if and only if $\exists M \in TM : f_M = f$. That is you only have to show existence of an appropriate Turing machine, not give one. What you -- we -- try to do there is to compute the Turing machine that computes the required function. This is a way harder problem!
The basic idea of the proof is: I give you an infinite class of functions, all of them computable (to show; trivial here). I prove then that the function you are looking for is in that class (to show; case distinction here). q.e.d.