I have two functions
$$g_1(n) = 1$$
$$g_2(n) = 10^{10^n}$$
I have to find one function that is neither $O(g_i(n))$ nor $Ω(g_i(n))$ ($i=1,2$). I already have : $$f(n) = 2 \sin (n)$$ That should be enough for $g_1(n)$, but I don't know how to come up with a function (or modify $f(n)$) so that it handles $g_2(n)$. I think that I should find a function that alternates near the $y$-axis.
Is there an easier way to do this?
Check that $$f(n)= 10^{(-11)^n}$$ works, and it has the advantage of being always positive.
EXPLANATION: In general, for positive functions $f,g$, $f(n) \in O(g(n))$ is equivalent on saying that $$f(n) \le C g(n)$$ i.e. $f(n)/g(n)$ is bounded. Similarly, $f(n) \in \Omega(g(n))$ means that $g(n)/f(n)$ is bounded.
$f (n) \in O(1)$ means that $f(n)$ is bounded
$f (n) \in O(10^{10^n})$ means that $f(n)/10^{10^n}$ is bounded
$f (n) \in \Omega(1)$ means that $1/f(n)$ is bounded
$f (n) \in \Omega(10^{10^n})$ means that $10^{10^n}/f(n)$ is bounded
Now, note that 1 implies 2 , and 4 implies 3. So, you want to find a function $f$ not satisfying 2 and 3.
Take a function such that even indices do not satisfy 2 and odd indices do not satisfy 3, and you are done.