For What Non-Negative Integer Values of $n$ is $n!\geq 3^n$

Question

How could I solve for the $n$ in this instance using discrete methods or is this something that I have to do by hand/computer? I've seen this problem in inductive proofs but the base case is usually given or calculated manually, is there a way to calculate using other methods?

Answer 1

$$2!<3^2$$ $$3!<3^3$$ $$4!<3^4$$ $$5!<3^5$$ $$6!<3^6$$ $$7!\geq 3^7$$ Afterwards, $\forall n\geq 8$, $$n!=(n)(n-1)...(9)(8)7!$$ $$\geq (3)(3)...(3)(3)7!$$ $$=3^{n-7}7!$$ $$\geq 3^{n-7}.3^7$$ $$=3^n$$

Hope it helps:)

Answer 2

You can find when the geometric mean of the LHS is greater-than or equal to that of the RHS. Since the RHS is constant in this regard, you want to know for which n,

$$\sqrt[n]{1\cdot 2\cdots n} \ge 3$$

One way to get an upper bound amounts to finding $k$ such that $1\cdot 2\cdot k\cdot (k+1)\ge3^4$. That is, you need two consecutive integers which more than "balance out" the geometric discrepancy with 3 from 1 and 2.

$$2k^2+2k-81 \ge 0$$

Which should be straight-forward. In this case the upper bound ends up being the correct $k$, but in general you may have a set of values to check.

Answer 3

Making the problem more general, at a point, you will need to solve for $n$ the equation $$n!=a^n$$ If you have a look at this question of mine, you will find a magnificent approximation proposed by @robjohn, an eminent MSE user. Making $k=0$, his approximation write $$\color{blue}{n\sim ea\exp\left(\operatorname{W}\left(-\frac{ \log(2\pi a)}{2ea}\right)\right)-\frac12}$$ where appears Lambert function.

For illustration, a few results $$\left( \begin{array}{ccc} a & \text{approximation} & \text{solution} \\ 1 & 0.92884 & 1.00000 \\ 2 & 3.44447 & 3.45987 \\ 3 & 6.00778 & 6.01602 \\ 4 & 8.60960 & 8.61515 \\ 5 & 11.2347 & 11.2389 \\ 6 & 13.8753 & 13.8787 \\ 7 & 16.5270 & 16.5297 \\ 8 & 19.1868 & 19.1892 \\ 9 & 21.8531 & 21.8552 \\ 10 & 24.5245 & 24.5264 \\ 11 & 27.2001 & 27.2018 \\ 12 & 29.8792 & 29.8807 \\ 13 & 32.5613 & 32.5627 \\ 14 & 35.2459 & 35.2471 \\ 15 & 37.9327 & 37.9339 \\ 16 & 40.6214 & 40.6225 \\ 17 & 43.3119 & 43.3129 \\ 18 & 46.0038 & 46.0048 \\ 19 & 48.6971 & 48.6980 \\ 20 & 51.3917 & 51.3925 \end{array} \right)$$

In the linked Wikipedia page, you will find series expansions for an approximation of $W(x)$ that you could use with a pocket calculator.