# Thread: Help with some function questions

1. ## Help with some function questions

Hey guys im revising for an exam and they have given us some extra revision questions as a challenge however i have no clue on how to do these 3 questions, if anyone can explain them to me or give me a solution so i can understand it would be helpful thanks

http://img176.imageshack.us/img176/9329/mathso.png

i cropped out the 3 questions i don't understand on the above picture, any help is appreciated thanks

2. 4. We have that $1+\log_2n<2\log_2n$ for $n>2$. Also, $\displaystyle\log_2n=\frac{\ln n}{\ln 2}$.

6. Note that $e < 3$ and $e^6<6!$. Therefore, $\displaystyle\frac{n!}{e^n}=\frac{6!\cdot7\cdot\ld ots\cdot n}{e^6\cdot e^{n-6}}>\frac{7}{e}\cdot\ldots\cdot\frac{n}{e}>\left(\ frac{7}{3}\right)^{n-6}$ for $n>6$.

For more help with these three problems, I would ask you to write the precise statements you need to prove by expanding the definition of big-O for these particular situations. Also, write what your difficulty is with proving those statements.

Edit: If I understand correctly, question 4 asks to show that $1+\log_2 n\in O(\ln n)$ directly using the definition of big-O.