Adela D. answered • 06/24/20
PhD Student in Computational & Applied Math with Tutoring Experience
Hi Sami! Good question: asymptotic bounds can be confusing at first.
In response to your question: no, there does not exist any constant k such that kn^4 > n^7 for all n sufficiently large. In order to see this, we can construct a proof by contradiction. If such a constant k (some real number) did exist, then it would have to satisfy k > n^3 for all n sufficiently large. However, by definition no constant real number can do this, because constants don't grow with n. In particular, for any real number k, I can find a value of m such that for all n larger than m, kn^4 < n^7.
These asymptotic bounds are captured by big-O notation. We write n^4 = O(n^7) precisely because n^7 dominates (i.e. is larger than) any constant multiple of n^4 in the limit as n goes to infinity. In fact, for any natural numbers (non-zero integers) p and q, if q < p then n^q = O(n^p).
