Difficulty proving / disproving the following equalities relations ( Big Ω)

Question

I have left with some functions I can't find witenesses for proving/disproving Big Ω equalities relations.

Here are the three relations:

1. $ \sum\limits_{i=1}^{n} (i^3 - i ^2) = \Omega(n^4) $
2. $log(n-\sqrt{n}) = \Omega(logn)$
3. $log(n+\sqrt{n}) = \Omega(logn)$

For the first one I tried to split the sum but to no avail. The second and the third - I don't even have an idea how to start proving/disproving this equality..

Thank you very much !

Answer 1

$$ S_1 = \sum_{k=1}^{n}(k^3-k^2)\geq \frac{n^4}{4}-\frac{n^3}{3} = \Omega (n^4) $$

The $\geq$ is due to comparison with the integral. $$ S_2=\log (n-\sqrt{n})=\log (\sqrt{n}(\sqrt{n}-1))\geq \frac{1}{2}\log n + \frac{1}{2} \log n - \log 2 \geq \log n -\log 2 = \Omega(\log n) $$ The first $\geq$ is due to $\log (\sqrt{n}-1) \geq \log \big( \frac{\sqrt{n}}{2} \big) $

can you handle the third one from here?