# Big O - proving that an estimate is correct

I recently submitted this for homework. The question asked to give a big-O estimate for (1) below. I have included the feedback in bold. It seems the solution I proposed lacked a proof and I unsure what I need to do to prove it. I wasn't even aware I had to offer a proof for a question that asked to just give a big - O estimate. Do I need to show how I arrived at (2) by including a statement like if (n) = O(a(n)) and g(n) = O(b(n)) then f(n).g(n)=O(a(n).b(n))? Please could someone advise what a complete solution should look like as I would like to correct my error

$$(1) \space(n! + 2^{n+3})(111n^3 + 15\log(n^{201} +1))$$

• $n! = O(n^{n})$ <<---"Better O(n!)"

• $2^{n+3}=O(2^{n+3})$

• $111n^{3}=O(n^{3})$

• $15\log(n^{201} +1)= O(15\log n^{201})$

Therefore the dominant term appears to derive from $(n!)\cdot(111n^{3})$ which would give us the following:

(2) $\space O(n^{n+3}) \space or \space O(n!n^3)$ . <<-- "You need to prove it"

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Refer to the Examples and see if this makes sense. Can you modify your work and put it in this form as a proof? –  Amzoti Dec 16 '12 at 14:42
cheers @Amzoti. –  bosra Dec 16 '12 at 14:44