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Questions tagged [alternative-proof]

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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Equivalence of all representations of $\exp$

I can name at least 4 different ways of representing $\exp$ function: Taylor series: For $x \in \mathbb{R}, \exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$. Differential equation: $f: \mathbb{R} \to \...
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Laurent series $f(z)=\dfrac{\sqrt{z}}{z+i}$

I came across to calculate an integral for which I had to find the Laurent series of $f(z)=\dfrac{\sqrt{z}}{z+i}$. I see that at $z=-i$ the function has a simple pole and the Laurent series I got is ...
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1answer
224 views
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Prove that, the function $f$ is injective: $f \big(f(x)f(y)\big) + f(x +y) = f(xy).$

I need to learn, by which method, I can prove that the function $f$ is injective. I would like to ask you to explain this problem using more detailed, more understandable, clearer and simpler ...
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1answer
37 views

Radius of convergence of power series $\displaystyle \sum_{n=1}^{\infty}\frac{x^{6n+2}}{(1+\frac{1}{n})^{n^2}}$

Find radius of convergence of $\displaystyle \sum_{n=1}^{\infty}\frac{x^{6n+2}}{(1+\frac{1}{n})^{n^2}}$. My try: Using Cauchy's Root test for convergence $\displaystyle \lim_nsup \sqrt[n]{|a_n|}|(z-...
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23 views

Is it possible to prove Fubini’s Theorem without Dynkin’s Theorem or the Monotone Class Theorem?

Fubini’s Theorem for Lebesgue integrals states that if $X$ and $Y$ are Sigma-finite measure spaces then the integral of a (well-behaved) function $f(x,y)$ with respect to the product measure on $X\...
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1answer
30 views

Limit of sequence $x_n$

For $\alpha \in (0,1)$ let the sequence $\{x_n\}$ be such that $x_0 = 0, x_1 = 1 $ and $x_{n+1} = \alpha x_n + (1-\alpha)x_{n-1},\quad n\geq1$. Find $\displaystyle \lim_{n\rightarrow \infty}x_n$. My ...
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1answer
22 views

Finding the unconditional distribution of a binomial distribution

This question is from a the textbook Mathematical Statistics by John Rice. I found a solution online for this question but am having a difficult time understanding the steps they took. In the line ...
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2answers
30 views

If the set of all upper bounds of $A$ and $B$ are equal and $\sup A$ exists, then $\sup B$ exists and $\sup A=\sup B$.

Caution: Axiom of Completeness is not assumed here. Before reading my attempt, I want you to think up a proof of your own. Here is my attempt: This concludes my proof. Two questions: How do I show ...
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1answer
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What is a formal full proof that “$ \mathbf{Set}$ is a Category” in the context of Category Theory?

I am learning category theory and couldn't find a full formal proof of the simplest example I could think of (that $\mathbf{Set}$ is a Category). What's seems the "hello world" of category theory? ...
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1answer
59 views

Proving $3x^{10} - y^{10} = 1991$ has no integral solutions. Check my proof.

The problem is Prove that $$3x^{10} - y^{10} = 1991$$ has no integral solutions. I have written this proof, of which I am not sure. Assume that $x$ and $y$ are integers. $$\begin{align} 3 x^{10}...
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1answer
51 views

Showing $x^4+x^2+x+1$ irreducible over $\mathbb{Z_3}$

$x^4+x^2+x+1$ irreducible over $\mathbb{Z_3}$. So since there are no roots there are no linear factors. From here do you just try to factor it as a product of 2 quadratics and show it that this leads ...
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2answers
118 views

Solutions of $n^{n}=2n$?

What is the best way to find solutions to $n^{n}$ and $2n$? Someone suggested the Lambert W function, but I cannot find a way to set it up in a way to use that. Any other suggestions? Meaning, what ...
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0answers
18 views

elementary question concerning cyclotomic polynomials

Is there a way to prove by induction (I can do it purely using the definition and a bit of playing around) that a polynomial in the form $z^{2^n}-1$ has a cyclotomic polynomial in the form $z^{2^{n-1}}...
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If $\frac {a}{a+bz+cz^2} = 1+p_1z+p_2z^2+…$ then $\frac {a+cz}{a-cz} \frac {a^2}{a^2-(b^2-2ac)z+c^2z^2} = 1+p_1^2z+p_2^2z^2+…$ [duplicate]

I can prove by binomial expansion that if $$\frac {a}{a+bz+cz^2} = 1+p_1z+p_2z^2+…$$ then $$\frac {a+cz}{a-cz} \frac {a^2}{a^2-(b^2-2ac)z+c^2z^2} = 1+p_1^2z+p_2^2z^2+...$$ But I wonder if there is ...
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5answers
56 views

Proving $5 \mid (n^5-n)$ for all $n \in \mathbb{Z}^+$

Prove for all $n \in \mathbb{Z}^+$ that $5 \mid (n^5-n)$ My proof Basis step: Since $5 \mid (1^5-1) \iff 5 \mid 0$ and $5 \mid 0$ is true, the statement is true for $n=1$. Inductive step: Assume ...
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3answers
81 views

Proving a series is convergent - $\sum _{n=1} ^\infty \frac{(-1)^n}{n}$ without using alternating series test

$$\sum _{k=1} ^\infty \frac{(-1)^k}{k}$$ I know this question has been answered a few times but my professor has not taught alternating series test yet or anything other than ratio test, root test ...
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2answers
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How to dissect a rectangle to obtain an equal square?

There are several propositions and constructions in Euclid's Elements that relate to the squaring of a rectangle, e.g. Proposition II.5: If a straight line is cut into equal and unequal segments, ...
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1answer
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If $f\mathop: X \to Y$ and $g\mathop: X \to Y$ are continuous and $f(x) = g(x)$ for all $x \in E$, then $f(x) = g(x)$ for all $x \in X$.

Let $X$ and $Y$ be metric spaces and $E$ be a dense subset of X. Show that if $f\mathop: X \to Y$ and $g\mathop: X \to Y$ are continuous and $f(x) = g(x)$ for all $x \in E$, then $f(x) = g(x)$ for ...
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4answers
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Is there a quick way to prove that $\sin(x)-x^2$ has exactly $2$ real roots?

I found a proof of this fact, but it is not concise as I would like. Would anyone have any advise on how to shorten it? Or an alternative shorter proof? Thank you very much in advance. $\textbf{Proof}...
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3answers
35 views

Can I derive the sum of squares formula without induction and through the formula for series?

I have $1^2+2^2+...+n^2$ and I want to prove the sum is $\frac{m(m+1)(2m+1)}{6}$. So for proving the formula for $1+3+5+7+... = n^2$ this is how i got the formula: the common difference is 2, so the ...
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1answer
21 views

Showing that set the of algebraic numbers are countable [Proof verification]

Consider the polynomial with (variable) integer coefficients : $ a_0x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$, where $x$ varies over the field of complex numbers. The set of all zeros of the ...
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0answers
25 views

Relationships between permutation and left representation of the same algebraic structure

A permutation group $P \leq S_n$ can be represented as a subgroup $H \leq GL(n,2)$ if a permutation $\sigma\in P$ acts in the indices of the vectors of $e_n$. Then every permutation of $P$ has a ...
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1answer
291 views

How integrating over a branch cut is made rigorous?

This is from Ch. 7 of the book Complex Variables by J Brown and R Churchill 8th ed. In evaluation of the counter integration of $f(z)=\dfrac{z^{-a}}{z+1}$ the book first suggests the following : ...
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47 views

Prove that $\hat\sigma^{2}$ is an unbiased estimator for $\sigma^{2}.$

Prove that $\hat\sigma^{2}$ is an unbiased estimator for $\sigma^{2}.$ Hint: You can use the result $E(x'Ax)=tr(A\sum)+\theta'A\theta$, where $E(x)=\theta$ and $V(x)=\sum.$ We want $E(\hat\sigma^2)=\...
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1answer
23 views

How to write that if $b_n = 0$ or $b_n = a_n$ and the serie of $a_n$ converges and is positive, then $b_n$ is also positive converges

I'm trying to find a rigorous and right way to prove my idea. If anyone could tell me how to do so that it's not too long but also makes sure that everything written is true, I would really appreciate ...
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0answers
27 views

Inequalities among the slopes of the sides and the slopes of internal angle bisectors

Suppose a triangle ABC without sides nor internal angle bisectors parallel to y-axis, and let $m_1, m_2, m_3 $ be the slopes of the equations of sides BC, CA, AB, and $s_1, s_2, s_3$ the slopes of ...
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0answers
16 views

Nice dimension independent proof of entropy inequality?

How does one prove that for arbitrary finite number of random variables $X,Y,Z,T,...$ the Shannon Entropy inequality holds $H(X,Y,Z,T,...) \leq H(X)+H(Y)+H(Z)+H(T)+...$ I know how to do it with ...
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2answers
32 views

Irreducible factors in an arithmetic sequence

The question is Let $T=\{1, 4, 7, 10, 13, 16, 19,...\}$. An element of $T$ is called irreducible if it is not $1$ and its only factors within $T$ are 1 and itself. Demonstrate that every element ...
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2answers
184 views

Comparing logarithms with different bases

$\log_3 4$ and $\log_7 10$: which of these two logarithms is greater? I figured out that both are between $1$ and $2$, then between $1$ and $1.5$. And then $\log_34$ is greater than $1.25$, and $\...
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1answer
16 views

Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest element of A.

We're proving both implications, so this is my first implication Proof: Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest ...
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2answers
203 views

Modelling a linear minimization program

Attempt: Let $x_l=$ number of barrel of light crude oil. $x_h=$ number of barrel of heavy crude oil. Then we should have as objective function $z=20x_l+15x_h$ And as conditions s.t. $.4x_l+....
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1answer
28 views

Generalisation of $A^{k}$ matrix

Question: If a $n$-by-$n$ matrix $A$ be $A=\begin{align} \begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \...
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3answers
64 views

Prove $\frac{2^{4n}-(-1)^n}{17} \in \mathbb{N}$ by induction

Here is my attempted proof: $\forall n \in \mathbb{N}$, let $S_n$ be the statement: $\frac{2^{4n}-(-1)^n}{17} \in \mathbb{N}$ Base case: $S_1$: $\frac{2^{4(1)}-(-1)^1}{17} = \frac{16+1}{17} = 1 \in \...
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2answers
71 views

The cyclic quadrilateral and the slopes of its sides

Suppose a plane quadrilateral ABCD (convex, concave or crossed) no side of which is parallel to y-axis, and let $m_1, m_2, m_3, m_4$ be the slopes of the equations of sides AB, BC, CD, DA. Having ...
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2answers
81 views

Show that $\int_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~dx~=~-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

Within this collocation of definite integrals number $30.$ is given by $$\int_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~dx~=~-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}\tag1$$ I figured out ...
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30 views

Prove that every triangle has a circumcenter with sets

Let $p=\{p_1,p_2,p_3\}\subset\Bbb R^2$ such that there does not exist a line $l\subseteq\Bbb R^2$ where $p\subset l$. Let $p_1=(a_1,b_1)$, $p_2=(a_2,b_2)$, $p_3=(a_3,b_3)$, where $p_1\not=p_2\not=p_3$...
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1answer
22 views

Maps of posets with $f(x)<g(x)$ induce homotopic functions

I have this question from a Quillen's paper Homotopy Properties of the Poset of Nontrivial p-Subgroups of a Group: Homotopy property: If $f,g\colon X\rightarrow Y$ are maps of posets such that $f(x)...
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2answers
52 views

Reformulating the definition of compactness

Consider the following alternative definition of a topological space. Definition: A topological space is an ordered pair $(X, \lessdot)$ consisting of a set $X$ and a binary relation $\lessdot$ ...
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172 views

Is this seemingly new proof that $[0, 1]$ is compact correct?

Hi all I think I found a new proof that $[0, 1]$ is compact but I am not 100% if it is correct, could you help me check? In the usual proof we just take the ...
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1answer
76 views

If $\sin\alpha+\sin\beta+\sin\gamma=\cos\alpha+\cos\beta+\cos\gamma=0$, then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$.

If $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$ Then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$. Try: $$(\sin \alpha+\sin \beta+\sin \...
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21 views

If $(x,y)=1 $ then $ (tx+y,n)=1$ [duplicate]

Prove that if $(x,y)=1$ than for all $n \in N$ exist $t \in N $ such that $$(tx+y,n)=1$$ I think this result is true because it follows from Dirichlet's theorem, although there must be an elementary ...
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2answers
78 views

$a + b = 2$ implies $a^c + b^c \ge 2$ for any real $c \ge 1$

If $a, b, c$ are positive reals such that $c \ge 1$ and $a + b \ge 2$ then $a^c + b^c \ge 2$. Is there any elementary way to prove it without using calculus and advanced inequalities like Jensen's? ...
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0answers
44 views

What's the fallacy in $i^{-1} = i$ [duplicate]

We need to express $i^{-1}$ as $a+bi$ where $a,b \in \mathbb{R}$. There are a lot of ways to simplify it. One can easily see that $i^{-1} = i^3= -i$, for example. However, this is not the approach ...
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2answers
34 views

We have $5$ red, $3$ blue and $2$ green balls. We take $3$ of them at random. What is the probability that they are of different colors?

We have $5$ red, $3$ blue and $2$ green balls (balls of the same color are indistinguishable). We take $3$ of them at random. What is the probability that they are of different colors? I have two ...
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0answers
36 views

Null homologous loop and orientable surface

I am reading Algebraic Topology: A First Course written by Greenberg and Harper. On page 67 of this book it is stated that Let $\gamma$ be a loop in $X$ regarded as a map $f:S^1\to X$. For $\chi[\...
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1answer
46 views

How to solve $x^2+y^2=pz^2$ in $x,y,z\in\mathbb{N}$ if $p$ is such a prime that $p\equiv 3 \pmod 4$?

How to solve $x^2+y^2=pz^2$ in $x,y,z\in\mathbb{N}$ if $p$ is such a prime that $p\equiv 3 \pmod 4$? Is the following proof OK? For such a prime we have lemma: $$p\mid a^2+b^2 \implies p\mid a\;\; {...
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3answers
80 views

Let $(P,<),(Q,\prec)$ be countable, dense, and linearly ordered sets without endpoints. Then $(P,<),(Q,\prec)$ are order-isomorphic

This theorem is usually proved constructively by Cantor's back-and-forth method. Is there other proofs for this well-known theorem? Especially, the proofs that don't define an explicit isomorphism ...
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1answer
21 views

Geometric proof that cross product is linear

I've been searching online quite a bit for a simple and elegant geometric proof that the cross product distributes over addition. That is, $$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \...
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2answers
40 views

Can the proof of finding the minimum of $x^2 + \frac{25}{x^2} + 3$ validly end with $x^2 + \frac{25}{x^2} + 3 \geq 13$?

I have a question where this is this expression needs to be found where it is the minimum: $x^2 + \frac{25}{x^2} + 3$ where $x > 0$ There are two options to solving this proof: (1) Using the ...
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2answers
41 views

Finding the required angle in the triangle

I have the following question with me: "Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC$ = $\angle ABC$ and $\angle BAM = 105^{\circ}$ . Find the measure ...