We know of course that rational+irrational=irrational and that ln(n) is irrational for every n greater than 1. Then why should $$\lim_{n\to \infty} \sum_{k=1}^n \frac{1}{k} - \ln(n)$$ not be irrational? Bear with me for a second, of course i know that there are counterexamples of such things, like $$x_n=\log(2)-\sum_{k=1}^n\frac{(-1)^{k+1}}k$$ But every counterexample i found looks at some series, where the underlying sequence goes to 0 and the other term is constant, or where both sequences tend to 0 like $$\frac{1}{n}+\frac{\pi}{n}$$. Or just take $\frac{\pi}{n}$, of course this is a sequence of irrationals which tends to a rational number. But these are not the exact cases for the definition of the Euler-Mascheroni constant. Both terms go to infinity. So is there a counterexample for this specific case too? Or has anyone ever looked into this specific case? Couldn't find anything on it. Maybe in the special case of $\infty-\infty$ one can indeed deduce a few things, if the limit exists.
So my exact Question is: Does there exist a rational number, which is the limit of a sum of rational numbers - a sequence of irrational numbers where both tend to infinity?
