A real number is called computable if there is an algorithm that computes it up to any desired precision. Suppose $f(x)$ is an elementary function s.t. $\sum_{n=1}^\infty f(n)$ converges (for simplicity let's say absolutely). Must the number $\sum_{n=1}^\infty f(n)$ be computable?
My understanding is that if we merely assume $f(n)$ to be a computable sequence of computable numbers then there are counterexamples, e.g., Specker sequence. On the other hand, if $f(x)$ is a rational function then this should be true, because we can bound the tail very explicitly. Is there a similar bound for arbitrary elementary functions (or at least some large class of functions)?