I was wondering if the asymptotic behavior of a divergent series' partial sums had anything to do with the regularized divergent series. For example, how does $\sum_{n=1}^\infty\sqrt{1+n^2}$ compare to $\sum_{n=1}^\infty n$, where we apply any method of regularization to both series? I note that zeta regularization is easy to do in the second, but not in the first, for example, but as for the partial sums, they are asymptotic:
$$\lim_{N\to\infty}\cfrac{\sum_{n=1}^N\sqrt{1+n^2}}{\sum_{n=1}^Nn}=1$$
I'd like to use the above as an example, but I'd still like answers to talk more generally about the relationships of regularization and asymptotic behavior of partial sums.