周期函数的渐近关系

Question

can n^(1 + sin n) be written as O(n^k) where k can be any positive integer greater than or equal to 2(k>=2)? And are asymptotic notations defined only for increasing functions with constant growth rate or they can be applied to wider range like decreasing function or periodic function? More insights about the same are very much welcomed.

Answer 1

Yes you can use asymptotic notation for periodic functions, but not for all.
The maximum value of sin(x) is 1 , and minimum value is -1 .

So we can say there's a subset of the natural numbers such that the restriction of f: n -> n ^{(1 + sin n)} to it is O(1)

Answer 2

You can use asymptotic relation for periodic functions.Here in your question n^(1 + sin n) = O(n^2) .
We can use f(n)=Θ(g(n)) means we can give both lower bound and upper bound to the function.

f(n)=Θ(g(n)) iff
f(n)<=c1.g(n)
f(n)>=c2.g(n)

where c1 and c2 are some constants.