How to calculate O-Notation? I am revising for my algorithms exam and I have come across one topic in particular that I do not quite understand;
What I would like to ask, if there is a certain way to find out O-Notation? Actually I know that the purpose of O-Notation, it helps us to estimate the effciency of algorithm. But in a specific questions like O(2^n) * O(n^10) I assumed that O(2^n * n^10) as in the worst case but learned that I was false, it should be 2^n and I don't know why? Shouldn't we think that which term would go faster to eternity, and in this case I thought that when I multiply both then it can go faster than 2^n, can't it? Could anyone explain please ? Just short and with beginner sight:)
We can show that $O(2^n\cdot n^{10}) \not\subset O(2^n)$:
As $2^n\cdot n^{10} \in O(2^n\cdot n^{10})$, then, if also holds $2^n\cdot n^{10} \in O(2^n)$ then we will have $2^n\cdot n^{10} \leqslant C 2^n$, which gives false $n^{10} \leqslant C$.