Thursday, 9 January 2020

real analysis - How to find $lim_{hrightarrow 0}frac{sin(ha)}{h}$

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How to find $\lim_{h\rightarrow 0}\frac{\sin(ha)}{h}$ without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...

summation - Equality of the sums $sumlimits_{v=0}^k frac{k^v}{v!}$ and $sumlimits_{v=0}^k frac{v^v (k-v)^{k-v}}{v!(k-v)!}$

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How can one proof the equality $$\sum\limits_{v=0}^k \frac{k^v}{v!}=\sum\limits_{v=0}^k \frac{v^v (k-v)^{k-v}}{v!(k-v)!}$$ for $k\in\mathbb{...

elementary number theory - How does one show that for $k in mathbb{Z_+},3mid2^{2^k} +5$ and $7mid2^{2^k} + 3, forall space k$ odd.

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For $k \in \mathbb{Z_+},3\mid2^{2^k} +5$ and $7\mid2^{2^k} + 3, \forall \space k$ odd. Firstly, $k \geq 1$ I can see induction is the best...

summation - How can you prove that $1+ 5+ 9 + cdots +(4n-3) = 2n^{2} - n$ without using induction?

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Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to kn...

calculus - What is wrong with treating $dfrac {dy}{dx}$ as a fraction?

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If you think about the limit definition of the derivative, $dy$ represents $$\lim_{h\rightarrow 0}\dfrac {f(x+h)-f(x)}{h}$$, and $dx...
Wednesday, 8 January 2020

real analysis - Sin(n) and cos(n) dense in $[-1,1]

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We knows that $sin(x)$ and $cos(x)$ are two function with value in the closed set $[-1,1]$ . How can I prove that $X=({sin(n)|n\in\mathbb{...

linear algebra - Given a Characteristic Polynomial of a Matrix...

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This question contains three parts. I have already answered the first two. The last part is confusing me. Suppose $A$ is a $4 \times 4$ ma...