Intereting Posts

Proving that these two sets are denumerable.
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
Show determinant of matrix is non-zero
Shorter proof for some equvalences
$\epsilon$-$\delta$ limits of functions question
Can we prove the existence of $A\cup B$ without the union axiom?
Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.
Compactness of the set of $n \times n$ orthogonal matrices
Explicit construction of Haar measure on a profinite group
Convergence of $\sum_n \frac{|\sin(n^2)|}{n}$
Examples of differentiable functions that are not of bounded variation
A game with two dice
Is this a valid use of l'Hospital's Rule? Can it be used recursively?
Proof of an inequality in a triangle
A question on $P$-space

How do I prove that $2^n=O(n!)$?

Is this a valid argument?

```
2<=2,
2<3,
2<4,....
2<n if n>2
therefore 2.2.2....n times < 1.2. ... n
so,2^n <n!
```

Thanks in advance.

- An issue with approximations of a recurrence sequence
- What are the rules for equals signs with big-O and little-o?
- How to prove that $\omega (n) = O\Big{(} \frac{\log(n)}{\log(\log(n))}\Big{)}$ as $n \to \infty$?
- Finding Big-O with Fractions
- Bounding a solution of an ODE with a small source
- Upper bound for $T(n) = T(n - 1) + T(n/2) + n$ with recursion-tree

- combinatorial question (sum of numbers)
- Probability question about married couples
- Coloring dots in a circle with no two consecutive dots being the same color
- 'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$
- Give a Combinatorial proof to show $\sum_{i=1}^{n}{iC(n,i)}=n2^{n-1}$
- How many combinations of $3$ natural numbers are there that add up to $30$?
- Counting number of moves on a grid
- Combinatorial proof for two identities
- Closed Form Expression of sum with binomial coefficient
- Number of certain (0,1)-matrices, Stanley's Enumerative Combinatorics

Since $e^2=\sum_{n=0}^{+\infty}\frac{2^n}{n!}$ converges (by ratio test if you want), the general term tends to $0$, whence, actually, $

2^n=o(n!)$ and not only $O(n!)$.

The argument is very informal and has a small hole in it, but the basic idea is correct. The hole lies in the fact that $2^n$ is actually larger than $n!$ for $n=1,2,3$. Properly you should show by induction on $n$ that $2^n\le n!$ for $n\ge 4$ and conclude immediately that $2^n$ is $O(n!)$.

$$\begin{align*}&(1)\;\;\text{Look at the positive series}\;\sum_{n=1}^\infty\frac{2^n}{n!}\\{}\\

&(2)\;\;\frac{a_{n+1}}{a_n}=\frac{2^{n+1}}{(n+1)!}\cdot\frac{n!}{2^n}=\frac2{n+1}\xrightarrow[n\to\infty]{}0 \;,\;\text{thus}\\{}\\

&(3)\;\;\text{The series in (1) converges}\\{}\\

&(4)\;\;a_n=\frac{2^n}{n!}\xrightarrow[n\to\infty]{}0\end{align*}$$

Thus, for some

$$N\in\Bbb N\;\;\text{and}\;\;\forall n>N\;,\;\;\frac{2^n}{n!}<1\implies 2^n=\mathcal O(n!)$$

You can use induction for $n > 3$.

Assume $ p(k): 2 ^ k < k!, k > 3 $

Prove that $p(k +1 ): 2 ^ {k + 1} < (k + 1)!$

$ 2 ^ {k + 1} = 2 ^ k * 2$

$ (k + 1)! = k! * k$

$ k > 2 \implies 2 ^ {k + 1} < (k + 1)! $

- Number of bitstrings with $000$ as substring
- Find the trajectories that follow drops of water on a given surface.
- Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$
- prove $x \mapsto x^2$ is continuous
- finitely generated k-algebra and polynomial ring
- If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
- Taking fractions $S^{-1}$ commutes with taking intersection
- Which functions are tempered distributions?
- Evenly Spaced Integer Topology is Metrizable
- Choice function for a collection of nonempty subsets of $\{0,1\}^\omega$
- Partition of ${1, 2, … , n}$ into subsets with equal sums.
- Compute the Jacobson radical of $R=\mathbb{F}_3$ and find all simple $R$-modules.
- Proving an expression is composite
- how do I prove that $\mathbb{Q} /\langle x^2 – 2 \rangle$ is a field
- Every point of a grid is colored in blue, red or green. How to prove there is a monochromatic rectangle?