**QUESTION**

Text

Image

2.) (20 points) Either by showing exact inequalities or using precise limits, find the asymptotic complexity of each of the following functions in the simplest terms that you can and then order them by "asymptotic dominance", i.e. produce an ordering $f_{1}(n) \ll f_{2}(n) \ll f_{3}(n) \ll \cdots$ where $g(n) \ll h(n)$ means that $g(n) \in O(h(n))$. So, for example, $1 \ll \log (n) \ll n \ll n^{2}$. Your ordering doesn't need justification. (a) $f_{a}(n)=\sum_{k=1}^{n^{2}}\left(\frac{3}{4}\right)^{k}$ (b) $f_{b}(n)=2 \log _{4}(4 n+17)$ (c) $f_{c}(n)=\sum_{j=1}^{2 n}(4 j+1)$ (d) $f_{d}(n)=6^{13}$ (e) $f_{e}(n)=5 n^{0.6}+3 n^{0.7}$ (f) $f_{f}(n)=\sqrt{3 n^{3}-2 n^{2}+7 n}$ (g) $f_{g}(n)=2 n \log _{3}\left(2 n^{3}+17 n+1\right)$*PLEASE SOLVE USING LIMIT NOTATION*