Are there theorems dealing with the "amount of oscillatory divergence" of series? Are there a set of theorems dealing with "amount of divergence" series?
Let me explain by example. The Dirchlet $\eta$ series $\sum_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say amount of divergences when $x>0$ is $0$. When $x=0$, the partial sum oscillates between $1$ and $0$, thus amount of divergence at $x=0$ is $1$ and more specifically the limit of oscillations is between $1$ and $0$. Are there theorems dealing with this kind of analysis?
I would like to examine if partial sum fluctuations stay within a certain range for various values of $x$ for a few series.
Thank you!
 A: The study of divergent series up to the early twentieth century was masterfully summarized in Hardy's book [1], see also [2]. Significant extensions were developed by Boshernitzan in the 1980's, see [3], [4], [5], with some overlapping work by Rosenlicht [6].
[1] Hardy, Godfrey Harold. Divergent series. Vol. 334. American Mathematical Soc., 2000.
[2] Tucciarone, J., 1973. The development of the theory of summable divergent series from 1880 to 1925. Archive for history of exact sciences, 10(1), pp.1-40.
[3] Boshernitzan, Michael. "An extension of Hardy’s class L of “orders of infinity”." Journal d’Analyse Mathématique 39, no. 1 (1981): 235-255.
[4] Boshernitzan, Michael. "Hardy fields and existence of transexponential functions." Aequationes mathematicae 30, no. 1 (1986): 258-280.
[5] Boshernitzan, M., 1984. Discrete" Orders of Infinity". American Journal of Mathematics, 106(5), pp.1147-1198.
[6] Rosenlicht, Maxwell. "Growth properties of functions in Hardy fields." Transactions of the American Mathematical Society 299, no. 1 (1987): 261-272.
