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 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?