# Asymptotics of a sinusoid

Consider the function $$f(n) = 2n^2 |\sin(\pi \cdot n/2)|.$$ Which of the following classes does $$f(n)$$ belong to? $$O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2).$$

I'm working in this and so far I have been trying to use different methods but always get to a dead end.

• The question title is misleading; the "sinusoid" is only an indicator function in disguise. – dkaeae May 13 '19 at 7:18

Let us notice the following: $$|\sin(\pi \cdot n/2)| = \begin{cases} 1 & \text{if n is odd}, \\ 0 & \text{if n is even}. \end{cases}$$ This implies that your function $$f(n)$$ alternates between $$n^2$$ (for odd $$n$$) and $$0$$ (for even $$n$$). This means, for example, that $$f(n) = O(n^2)$$ (since $$f(n) \leq n^2$$ for all $$n$$) but $$f(n) \neq \Omega(n^2)$$ (considering even $$n$$).