# Math Help - 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.

(the question is just like that they dont give a definition for the function O)

4. As a general rule, it is useless to try to solve problems without knowing the definitions of concepts involved. Mathematics is notorious for redefining common words to mean something special. For examples, phrases like "dominates", "tends to", "almost everywhere" have very precise meanings. It also helps to see several examples of how new concepts are applied to gets some intuition about them.

It is natural that the problems don't provide the definition of big-O. I would recommend reading about it in a textbook or online. See, for example, the Wikipedia article and the PDF link at the bottom of it or Google for "big-O notation".

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.