little o notation with natural logs

I'm having trouble with little o notation.

Help me show that:

$2(n^2 + 100n)\log^5n = o(n^2\sqrt{n})$.

It is the last hwk on my sheet and I don't understand it, if someone can help me with little o notation, that would be great thank you kindly.

• Have you been given a definition of the little-oh notation? – Gerry Myerson Oct 8 '12 at 22:53
• Can you show that $\lim_{n \to \infty} \frac{n}{\log n}= \infty$? – Alex Oct 9 '12 at 2:03

Hint: For positive functions $f(n)$ and $g(n)$, to show that $f(n)=o(g(n))$, one needs to show that $$\lim_{n\to\infty} \frac{f(n)}{g(n)}=0.$$ In our particular case, after writing down what $\dfrac{f(n)}{g(n)}$ is, I would suggest dividing top and bottom by $n^2$. You will get something like $$2\left(1+\frac{100}{n}\right) \frac{\log^5n}{n^{1/2}}.$$ Finally, you need information on the speed of growth of the logarithm, and powers of it, in comparison with the speed of growth of positive powers of $n$. It might help to note that $$\frac{\log^5 n}{n^{1/2}}=\left(\frac{\log n}{n^{1/10}}\right)^5.$$