Question concerning big-Oh and small-Oh notation

Question

What would the notation

$a_n = (1+ o(1))b_n$

stand for? (And similarly for $a_n = (1 + O(1))b_n$).

Answer 1

In the first case, what you have is that $a_{n}=b_{n}+o(1)b_{n}$. In the limit, any term that is $o(1)$ goes to zero, so you can think of the $b_{n}o(1)$ term as "lower order terms." This essentially says that $$a_{n}=b_{n}+\text{lower order terms}.$$ You can also think of this as saying that $a_{n}$ and $b_{n}$ are asymptotically the same.

The latter is rarely used since $1+O(1)=O(1)$ and $a_{n}=O(1)b_{n}$ means that $a_{n}=O(b_{n})$. You can think of $O(1)$ as just being "bounded by a constant." In this case that means that (for large enough $n$), $a_{n}\leq (1+C)b_{n}$ or $a_{n}\leq C'b_{n}$.