Answer by WHLin for How can i show that $\frac{k^3}{3^k}$ is monotonic...
Let $a_k \equiv \frac{k^3}{3^k}$.$a_{k+1}/a_{k} < 1$$\iff$$ \frac{1}{3}(1+\frac{1}{n})^3 <1$$\iff n>2.2614$
View ArticleAnswer by WHLin for How to find a solution of this equation system.
This is just some of my thought: Suppose $\gamma = v_1/v_2$ is finite, we have $ x^T(A-\gamma B)y = 0$. Since $x,y$ are nonnegative and nonzero, at least the matrix $A-\gamma B$ cannot be positive for...
View ArticleAnswer by WHLin for how to prove that Fibonacci sequence is divergent
We prove that $F_n \geq n-2$ by induction.(1) The inequality hold for $n=1,2,3,4$.(2) Suppose it is true for $F_n$ with $n=1,2,...,k+1$ for $k\geq3$, then we have $F_{k+2}=F_{k+1}+F_k \geq (k-1)+(k-2)...
View ArticleAnswer by WHLin for Is this weird function with argument in the integrand...
I think in general the statement is "false". Let $I = [0,2]$, $f: I^2 \rightarrow \mathbb{R}$, and $t_0 = 0$. Define$$ f(x,t) \equiv 0, \;\; \text{if } x \in [0,1) \cup (1,2] $$$$ f(x,t) \equiv 1, \;\;...
View ArticleAnswer by WHLin for A question about determinants an multilinear algebra
This is just some of my thought.Since the dimension is finite, we could represent $\phi$ and $\varphi$ as matrices $A$ and $B$ and consider $F$ as a matrix function mapping from $\Gamma^{n^2}$ to...
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Algebraic equation of a specific six-degree polynomial
During my research, I was analyzing a linear system with characteristic polynomial of six order, $p(x)=x^6+3ax^5+3a^2x^4+a^3x^3 - a^3b_1b_3b_3$. I used Mathematica and it found all roots as the...
View ArticleAnswer by WHLin for Why is $ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$...
By definition, we have $sin(t)=t-\frac{t^3}{3!}+\frac{t^5}{5!}-...$ and hence $sin(xyz)=xyz-\frac{(xyz)^3}{3!}+\frac{(xyz)^5}{5!}-...$Therefore, when $(x,y,z)$ approaches to zero, the behavior of...
View ArticleAnswer by WHLin for How do we show that $\lim_{x\to0}(1+x)^{\frac{1}{x}}$ is...
Let $f(x)\equiv (1+x)^{1/x}$, we have $f(x)=exp\{log[(1+x)^{1/x}]\}$. Since $exp\{.\}$ is continuous, we have$$\lim_{x\rightarrow 0} f(x)= exp\{\lim_{x\rightarrow 0} log[(1+x)^{1/x}]\}=...
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