Can anyone give a simple example of a sequence that converges, but there's no computable function that gives $N$ as a function of $\epsilon$, i.e., the [modulus of convergence](http://en.wikipedia.org/wiki/Modulus_of_convergence) is not computable?


In the literature, all I could find were aesthetically unpleasant examples of Specker sequences. 


I hope that relaxing the requirements of the sequence itself being computable and it's limit not is enough to get simpler examples. Unfortunately, the examples that I've been able to come up with myself are worse than the literature.




---


And what if we still require the sequence itself to be computable? Is there still so easy an example?



Simple example of a sequence without computable modulus of convergence
Because of the relaxed requirements, the following seems to work. Let $f$ be an increasing function from natural numbers to natural numbers that grows faster than any computable function. (For example, define $f(n)$ to be $n$ plus the largest number obtainable by giving any input $\leq n$ to any Turing machine with Gödel number $\leq n$.) Then define $a\_k$ to be $1/n$ if $f(n-1)<k\leq f(n)$. The sequence converges to 0, but any modulus of convergence, applied to $\varepsilon=1/n$, would majorize $f$ and would therefore not be computable. 


