Clarification on Term Order of an Infinitesimal

Question

I've seen several questions posted on this, namely

1. What is the order of an infinitesimal?

2. Determine the order of an infinitesimal.

I'm asking this question because I've seen it pop up in variational calculus and want to make sure the answers in the above posts apply.

e.g. Gelfand & Fomin's Calculus of Variations states (Section 3.2, pg. 11), for

$$
\begin{align}
\Delta J[h] &= J[y+h] – J[y] \\
&=\varphi [h] + \epsilon \lVert h \rVert,
\end{align}
$$

that "the linear functional $\varphi [h]$ which differs from $\Delta J[h]$ by an infinitesimal of order higher than 1 relative to $\lVert h \rVert$, is called the [first] variation of $J[y]$ and is denoted by $\delta J[h]$".

This kind of terminology appears more than once.

Question

Is the order of an infinitesimal, $\epsilon$, its integer power, $n$, such that

$$
\epsilon^n
$$

is an nth-order infinitesimal, with the only difference between orders of infinitesimals being how fast they converge to $0$?

(This definition being purely due to Cauchy, explained in his text, the English annotated version of which is given here:

http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/CauchyCoursdAnalyseAnAnnotatedTranslationSourcesandStudiesintheHistoryofMathematicsandPhysicalSciences.pdf))

Answer 1

Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.