I've been working my way through Introduction to Algorithms, Fourth Edition and am having a hard time with one of the exercises in chapter 3, which is mostly about standard notation.
The exercise is as follows:
3.3-3
Use equation (3.14) or other means to show that $(n + o(n))^k = Θ(n^k)$ for any real constant $k$. Conclude that $⌈n⌉^k = Θ(n^k)$ and $⌊n⌋^k = Θ(n^k)$.
Equation (3.14) is as follows:
$1 + x ≤ e^x$
I was unable to make a connection between this equation and the exercise, but reasoning about the properties of the notation, I think that when $n$ is sufficiently large, the $o(n)$ in $n + o(n)$ is relatively insignificant compared to $n$, and so the part inside the brackets is just $n^k$, which is a member of $Θ(n^k)$.
For $⌈n⌉^k$ and $⌊n⌋^k$ I'm really not sure what the exercise wants me to think about, because any $n^k$ is $Θ(n^k)$.
I would really appreciate a hint as to how I could repeat the exercise as intended, using equation (3.14), along with any other comments about the above.