Arrange in increasing order of asymptotic complexity

Question

I have the following functions that I need to rank in increasing order of Big-O complexity:

$$(\log n)^3, 10\sqrt n, n\log n, n\sqrt n, n^4 + n^3, (2.1)^n \cdot n^2, 3^n, 2^n \cdot n^3, n! + n, n^n. $$

My current ranking is as follows:

$$(\log n)^3 < 10\sqrt n < n\log n < n\sqrt n < n^4 + n^3 < (2.1)^n \cdot n^2 < 2^n \cdot n^3 < 3^n < n! + n < n^n. $$

Is my ranking of the functions in increasing order of complexity correct?

Answer 1

You have mistake in $(2.1)^n \cdot n^2<2^n \cdot n^3$, because it is equivalent $\left(\frac{2.1}{2}\right)^n<n$