Intereting Posts

Gauss Bonnet theorem validation with hyperbolic circles
Generalizing Ramanujan's cube roots of cubic roots identities
For what values of $x$ is $\cos x$ transcendental?
Why can't reachability be expressed in first order logic?
Find the value of $\space\large i^{i^i}$?
Bijection from (0,1) to [0,1)
Two different expansions of $\frac{z}{1-z}$
Evaluating the time average over energy
Find $\det X$ if $8GX=XX^T$
Average distance between two points in a circular disk
Asymptotic (divergent) series
The composition of two convex functions is convex
Show that $S$ is a group if and only if $aS=S=Sa$.
Is the tangent bundle the DISJOINT union of tangent spaces?
In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$.

I can do this myself, but can’t figure out any kind of “beautiful” way to do it.

**The way I do it is:**

- Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$
- Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$
- 'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?
- Identity concerning complete elliptic integrals
- Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?
- Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Assume $3^n \ge n^3$

Now,

$(n+1)^3 = n^3 + 3n^2 + 3n + 1$,

and $\forall{} n \ge 3$,

$3n^2 \le n^3, \,\, 3n + 1 \le n^3$

Which finally gives $(n+1)^3 \le 3n^3 \le 3^{n+1}$ by our assumption.

Now just test by hand for n=1,2,3 and the rest follows by induction.

**Anyone got anything simpler?**

- elementary prove thru induction - dumb stumbling
- Next step to take to reach the contradiction?
- The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer
- Prove that a continuous function on a closed interval attains a maximum
- Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
- Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
- Proof of Product Rule for Derivatives using Proof by Induction
- Alternative proof of $\sqrt{2}$ is irrational assistance.
- Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
- Flawed proof that all positive integers are equal

**Basis:**

$n = 1$

$$3^1 \ge 1^3 \implies 3 \ge 1\text{, which is true}$$

**Inductive hypothesis**

Let $n=k$ and also let this inequality hold:

$$3^k \ge k^3$$

**Inductive step**

We’ll prove that it also holds when $n=k+1$

$$3^{k+1} \ge (k+1)^3$$

$$3^k \cdot 3 \ge k^3 + 3k^2 + 3k + 1$$

$$3^k + 3^k + 3^k \ge k^3 + 3k^2 + 3k + 1$$

$$2 \cdot 3^k \ge 3k^2 + 3k + 1$$

Note that for every $k \ge 2$, this inequality holds $3k^2 \ge 3k + 1$

$$2 \cdot 3^k \ge 6k^2$$

$$3^k \ge 3k^2$$

This is true for every $k \ge 3$

Now you can prove the other 2 cases $n = 1,2$ to give a complete proof.

I don’t know about beauty, but after verifying the inequality for the first three cases, one can use the fact that $64\lt81$ to conclude that $3\log(1+{1\over3})\lt\log3$. By combining this with the induction hypothesis, it follows that for $n\ge3$,

$$\begin{align}

3\log(n+1) &= 3\log n + 3\log(1+{1\over n})\cr

&\le 3\log n +3\log(1+{1\over3}) \cr

&\lt n\log3+\log3\cr

&=(n+1)\log3

\end{align}$$

Here is another argument, but it’s not necessarily simpler than yours:

1) If $n=1$, $3^1=3\ge1=1^3$; if $n=2$, $3^2=9\ge8=2^3$; and if $n=3$, $3^3=3^3$.

2) Now assume that $3^n\ge n^3$ for some integer $n\ge3$.

Then $\displaystyle\frac{1}{n}\le\frac{1}{3}$, so $\displaystyle\frac{(n+1)^3}{n^3}=\big(1+\frac{1}{n}\big)^3\le\big(\frac{4}{3}\big)^3=\frac{64}{27}\le3$, so

$\;\;\;\;\;\;\;\;3^{n+1}=3(3^n)\ge3n^3\ge(n+1)^3$.

Show true for $n=1,2,3$, and assume true for $k$. Then note that $3^{k+1} = 3^{k}\cdot 3 \geq 3\cdot k^3 = (3^{1/3}\cdot k)^3$ (by our assumption). Now, for $k\geq 3$, we see that $3^{1/3}\cdot k \geq k+1$ and the result follows.

- Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.
- Is complex analysis more “real” than real analysis?
- Example for fintely additive but not countably additive probability measure
- Equality of limits on $\varepsilon – \delta$ proof
- How to calculate $\lim\limits_{{\rho}\rightarrow 0^+}\frac{\log{(1-(a^{-\rho}+b^{-\rho}-(ab)^{-\rho}))}}{\log{\rho}} $ with $a>1$ and $b>1$?
- $f(x)=x^m+1$ is irreducible in $\mathbb{Q}$ if only if $m=2^n$.
- How to show that $\gcd(n! + 1, (n + 1)! + 1) \mid n$?
- Proving that the number of vertices of odd degree in any graph G is even
- Is my understanding of product sigma algebra (or topology) correct?
- To prove that $ + ++\dots ++$ is even.
- Understanding Fatou's lemma
- Number of horse races to determine the top three out of 25 horses
- How to prove that an operator is compact?
- Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$
- Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces