divergent series and natural log difference can be rational?

Question

EDIT1:

Apart from harmonic series is there another divergent series whose limiting difference with natural logarithm is rational?

Answer 1

First of all, if a series diverges, then it doesn't really have a limit (unless you'd say $\infty$ is a limit).

If you on the other hand mean having a series $\sum a_n$ and $\sum b_n$, with difference $\sum (a_n-b_n)$, then sure: take $a_n=1+\tfrac1{2^n}$ and $b_n=1$ so that

$$\sum_{n=1}^{\infty}(a_n-b_n)=1$$

is rational, yet both $\sum a_n$ and $\sum b_n$ diverge.