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][1] 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.

  [1]: http://en.wikipedia.org/wiki/Modulus_of_convergence