For a sequence $\{\epsilon_n\}$ that approaches $0$ as $n\to\infty$, we say that it approaches zero "geometrically" if $$\lim_{n\to\infty} \frac{\epsilon_n}{\epsilon_{n-1}} = \lambda$$ for some $\lambda$ with $|\lambda < 1|$. Thus for example the sequence $2^{-n}$ approaches zero geometrically. For a sequence that approaches zero geometrically, the number of zeros in the decimal expansion of $\epsilon_n$ grows linearly with $n$.
I have also heard the term "approaches zero exponentially (fast)" applied to sequences where $$\lim_{n\to\infty} \frac{\epsilon_n}{\epsilon_{n-1}^\alpha} = \rho$$ for some $\alpha > 1$ and some finite non-zero $\rho$. Thus for example, if $s_n$ is the value obtained by the $n$-th iteration of Newton's algorithm in calculating $\sqrt{2}$ starting with any iniital guess $s_0 \in \Bbb{N}$, and $\epsilon_n = \sqrt{2}-s_n$, then the sequence $\{\epsilon_n\}$ approaches zero exponentially fast (as the error in Newton for the square root falls like $$ \epsilon_{n+1} = \frac{\epsilon_n^2}{2(\sqrt{2}+\epsilon_n)}\implies \lim_{n\to\infty} \frac{\epsilon_n}{\epsilon_{n-1}^2} = \frac{\sqrt{2}}{4} $$ For sequence that approaches zero exponentially, the number of zeros in the decimal expansion of $\epsilon_n$ grows geometrically with $n$.
My question is, what do we call a sequence that approaches zero "like $\frac1n$", that is, where $$ \lim_{n\to\infty} n\epsilon_n = \gamma $$ Or to be more general, when for some positive power $p$, $$ \lim_{n\to\infty} n^p \epsilon_n = \gamma $$ For $p=1$ I might call this approaching zero "harmonically" but I'm not sure if that is accepted terminology.