I. $[70$ pts$]$ Find the big $O$ for the following functions. $[$Notice that this problem requires you to find the most significant item in a function and that is the least upper bound of the function, and it is the big $O$ we look for. For example, $f(n)=3n+n^2+nlogn.f(n)=O(n^2),O(n^3),O(n^4),$dots The answer we want is $O(n^2).$ Formulas for your reference:

${\displaystyle \underset{i=1}{\overset{n}{}}}i=\frac{(1+n)n}{2}$

${\displaystyle \underset{i=0}{\overset{n}{}}}{a}^i=a^0+a^1+a^2+$dots$+a^n{?}_q=a^0{a}^0=a^{\text{l}\text{o}\text{n}\text{g}\text{r}\text{i}\text{g}\text{h}\text{t}\text{a}\text{r}\text{r}\text{o}\text{w}}=\frac{a^0-a^{n-1}}{1-q}$

${\displaystyle \underset{k=0}{\overset{n}{}}}{k}^2=\frac{n(n+1)(2n+1)}{6}$

${\displaystyle \underset{k=0}{\overset{n}{}}}{k}^3=0^3+1^3+2^3+$dots$+n^3=\frac{n^2(n+1{)}^2}{4}]$

$f(n)=8\sqrt[3]{n^7}+3n^2-n^6$

$g(n)=log\sqrt[\phantom{2}]{n}+loglogn$

$h(n)=9(4^{logn})+nlogn,[$simplify the $4^{logn}]$

$j(n)=log{n}^n+n^{log5}$

${\displaystyle \underset{k=0}{\overset{n}{}}}2k$

${\displaystyle \underset{i=0}{\overset{n}{}}}{2}^{\text{'}}$

${\displaystyle \underset{k=0}{\overset{n}{}}}{k}^2$

${\displaystyle \underset{k=0}{\overset{n}{}}}{k}^3$