series of rationals that converges towards two different rationals
We can convert between a rational series and rational sequence by $$b_n=\sum_{k=0}^{n-1} a_k$$ $$a_k=b_{n+1}-b_n$$
Now you can focus on finding just a sequence that converges to two different numbers. A very useful choice is to find a sequence that goes to $0$ in one and $1$ in the other field, here is one such example,
$$b_n = \frac{n!}{1+n!}$$
This goes to $0$ in every p-adic field and $1$ in the reals. So now it's a simple matter of picking any two rational numbers you like, $r_1,r_2$ and now make,
$$c_n = r_1b_n+r_2(1-b_n)$$
Now this converges to your choices.