Little-oh vs. $\ll$ notation

Question

Let $a_n$ and $b_n$ be two sequences. I'm trying to understand the difference between $a_n = o(b_n)$ and $a_n \ll b_n$.

$a_n = o(b_n)$ as $n \to \infty$ if $a_n/b_n \to 0.$

$a_n \ll b_n$ if $a_n \ge 0$ and $a_n = o(b_n).$

What is the importance of the requirement that $a_n$ be nonnegative in the second definition?

Answer 1

Little-oh notation just checks that $a_n$ grows at a slower rate than $b_n$. For instance, $a_n = -n$ and $b_n = n^2$, then $a_n = o(b_n)$. The $a_n \geq 0$ is supposed to mean that $b_n$ eventually is greater than $a_n$. Here, "greater than" refers to the total order on the sequence's values, which is not mentioned in the first definition.

Still, that definition seems incomplete to me. We wouldn't intuitively say $a_n \ll b_n$ if $a_n = n$ and $b_n = -n^2$, yet that fulfills the given definition for "$\ll$".

Answer 2

The standard definition of $a_n\ll b_n$ is equivalent to $a_n=O(b_n)$. That is, $|a_n|\leq C|b_n|$ for some constant $C>0$ and all sufficiently large $n$. So, for example, if we put $a_n=n\sin n$, we have $a_n\ll n$ since $|a_n|\leq 1\cdot n$, but we don't have $a_n=o(n)$.