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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32 views

Strong Induction proof : $x_k = 1/2(x_{k+1}+x_{k-1})-1$ holds for all integers $k∈Z_{≥0}$

I have just started to learn how to do strong induction. I am struggling to fully understand how to properly do it though. I am trying to work on this exercise where I must prove $x_k = 1/2(x_{k+1}+x_{...
3
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2answers
72 views

Find all real numbers $x$ such that $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number

I want to find all values of $x\in \mathbb R$ such that the value of $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number. I solved it as follows: $x+2\sqrt{x}-1\ge 0$ $(\sqrt{x}+1)^2-2\ge 0$ $...
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1answer
30 views

Let $U \subset S^n$ open subset of the sphere homeomorphic to $\mathbb{R}^n$ prove that $\mathbb{S}^n \setminus U$ is connected

Let $U\subset S^n$ open subset of the sphere homeomorphic to $\mathbb{R}^n$ prove that $\mathbb{S}^n \setminus U$ is connected Assume that $\mathbb{S}^n \setminus U$ isn´t connected then there ...
2
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1answer
53 views

Alternative proof of the Fibonacci property $\sum\limits_{j=n}^{n+9}F_j=11\cdot F_{n+6}.$

If we consider the standard Fibonacci sequence$$F_0=0;\quad F_1=1;\quad F_n=F_{n-1}+F_{n-2}\quad\forall n\geq 2,$$ it is easy to proof that, given any chunk of $10$ consecutive Fibonacci, its sum is ...
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1answer
72 views

Prove that $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{x\cos 2x\cos x}{\sin^{7}x}{\rm d}x= -\frac{16}{45}$ [closed]

Prove that $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{x\cos 2x\cos x}{\sin^{7}x}{\rm d}x= -\frac{16}{45}$$ The number $-\frac{16}{45}$ is quite well-known, like $\int_{a}^{b}f\left ( x \right ){\rm ...
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0answers
91 views

Understand if my proof is fallacious.

I would like to know if my attempted solution to problem 4 of chapter 3.5 from Velleman's book How to Prove It is valid. My solution is different from the one suggested by the book and I want to know ...
3
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2answers
143 views

Interesting logarithmic inequality

Recently the following question was published on the site and soon after this deleted: Prove the inequality $$ \left(\log\frac{a^2+b^2}{2ab}\right)^n\le \left(\log\frac{a^2+c^2}{2ac}\right)^n+\left(\...
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3answers
74 views

Calculation in Routh's theorem

The proof of Routh's theorem concludes with showing$$1-\frac{x}{zx+x+1}-\frac{y}{xy+y+1}-\frac{z}{yz+z+1}=\frac{(xyz-1)^2}{(xz+x+1)(xy+y+1)(yz+z+1)}.$$I seek an elegant "proof from the book" ...
5
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0answers
98 views

About proving that there are infinitely many prime numbers $p$ such that $\mathrm{ord}_p(a)=\mathrm{ord}_p(b)$

I have seen this question here, however, what interested me the most is the partial answer which uses $Fermat's \ little \ theorem$, but as you can see there, he didn't continue proving that the power ...
1
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1answer
50 views

Change the order of integration $\int_{1}^{\frac{4}{3}}{\rm d}x\int_{\sqrt{\frac{x}{3}}}^{2- x}f\left ( x, y \right ){\rm d}y$

Change the order of integration $$\int_{0}^{1}{\rm d}x\int_{\sqrt{\frac{x}{3}}}^{\sqrt{x}}f\left ( x, y \right ){\rm d}y+ \int_{1}^{\frac{4}{3}}{\rm d}x\int_{\sqrt{\frac{x}{3}}}^{2- x}f\left ( x, y \...
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4answers
92 views

Is $\tan(\alpha/2)=(1-\cos\alpha)/\sin \alpha, \,?$ [duplicate]

Is it true that $$\bbox[5px,border:2px solid #138D75]{\tan\left(\frac {\alpha}2\right)=\frac{1-\cos\alpha}{\sin \alpha}, \quad ?} \tag1$$ My solution: First, we note that the expression $\tan \left ( \...
4
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2answers
133 views

To need a way of thinking about Ji_Chen's nice result $(a- c)^{2}+ (b- d)^{2}\geq\frac{7}{9}ab- \frac{7}{20}(c^{2}+ 4d^{2})$

given four real numbers $a, b, c, d$ Ji Chen gave a nice result on.AoPS $$\left ( a- c \right )^{2}+ \left ( b- d \right )^{2}\geq\frac{7}{9}ab- \frac{7}{20}\left ( c^{2}+ 4d^{2} \right )$$ The ...
0
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1answer
40 views

Trying to prove by induction that $\sum_{i=1}^{k} F_{2i}=F_{2k+1}-1$ for all $k∈N$ [duplicate]

I am trying to prove by induction the following proposition of the fibonacci sequence: The fibonacci sequence is defined recursively as: $f_1 = 1$, $f_2 = 1$, and that for all integers $k>=1$, $f_{...
2
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1answer
33 views

If $\tan x\tan y=\frac{b}{a},\,a,\,b\ne 0$, prove that $\frac{\sec^2x}{a\tan^2x+b}+\frac{\sec^2y}{a\tan^2y+b}=\frac{a+b}{ab}$

If $\tan x\tan y=\frac{b}{a},\,a,\,b\ne 0$, prove that $\frac{\sec^2x}{a\tan^2x+b}+\frac{\sec^2y}{a\tan^2y+b}=\frac{a+b}{ab}$ I solved it in the following way: $\frac{1+\tan^2x}{a\tan^2x+b}+\frac{1+\...
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3answers
70 views

Solve in the set of integers $2^x +5^x = 3^x + 4^x$

Find the number of integer solutions(both positive and negative) of the equation: $$2^x +5^x = 3^x + 4^x$$ With induction we see that for $x\geq 2$ we have $$5^x\geq 3^x+4^x$$ It is trivially true ...
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0answers
23 views

Proving gradient descent is a contraction for strongly convex functions

I am trying to show that the gradient mapping $G: x \to x-\eta \nabla f(x)$ of an $L$-smooth function $f$ is contraction iff $f$ is $\mu$-strongly convex. It is easy to show that for an $L$-smooth and ...
1
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2answers
33 views

Prove $F_n=(-1)^{n+1}F_{-n}$ without induction

Consider the Fibonacci sequence defined by $$F_{n+2}=F_{n+1}+F_n,~~~~~~F_1=F_2=1$$ I can prove via induction that $F_n=(-1)^{n+1}F_{-n}$, but how can it be proven without using induction? Thank you ...
1
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2answers
37 views

Evaluating B(3/4,5/4) without the Legendre duplication formula?

Suppose that one wanted to evaluate $B(3/4,5/4)$ where $B$ is the beta function. One approach is as follows. $$B(3/4,5/4)=\frac{\Gamma(3/4)\Gamma(5/4)}{\Gamma(3/4 + 5/4)}=\Gamma(3/4)\Gamma(5/4).$$ ...
3
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1answer
47 views

Given a recursion $\frac{a_{n+1}}{4a_{n+1}+n+1}=\frac{2(a_n+n)^2}{9n(4a_n+n)}$ with $a_1=1.$ Show that $a_n\sim\frac n2\,{\rm as}\,n\rightarrow\infty$

Given a recursion $\dfrac{a_{n+ 1}}{4a_{n+ 1}+ n+ 1}= \dfrac{2\left ( a_{n}+ n \right )^{2}}{9n\left ( 4a_{n}+ n \right )}$ with $a_{1}= 1.$ Prove that $$a_{n}\sim\frac{n}{2}\,{\rm as}\,n\rightarrow\...
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1answer
47 views

Given a recursion $a_{n+1}=\sqrt{a_n^2+a_n}$ with $a_1=1.$ Prove that $\lim{\left(a_n\right)}'=\frac12$ without using $a_n\sim\frac n2-\frac14\ln n$

Given a recursion $a_{n+ 1}= \sqrt{a_{n}^{2}+ a_{n}}$ with $a_{1}= 1.$ Prove that $\lim{\left ( a_{n} \right )}'= \frac{1}{2}$ without using the result I've got $$a_{n}\sim\frac{n}{2}- \frac{1}{4}\ln ...
2
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1answer
51 views

Prove that $a ≤ x ≤ b ⇒ |x| ≤ |a|+|b|$

Can you please help me with this proof $$a ≤ x ≤ b ⇒ |x| ≤ |a|+|b|$$ ? I am literally stuck for hours. This is what i thought, but i don't know if it counts as a proof. First of all, if a ≤ x then $x ...
4
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0answers
152 views

Given three natural numbers $x, y, z.$ Prove that $xy+ yz+ zx\neq 462$ and $xyz+ x+ y+ z\neq 1193$

Given three natural numbers $x, y, z.$ Prove that Problem 1. $$xy+ yz+ zx\neq 462$$ without loss of generality, I accept $x:=\min\left \{ x, y, z \right \}\Rightarrow 3x^{2}\leq 462\Rightarrow x\leq ...
0
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0answers
58 views

Algebraic complex numbers are countable without using fundamental theorem of algebra

I want to prove that the set of all algebraic numbers is countable. I know the proof where we show that the set of all polynomials with integer coefficients is countable (it is the union of $n$ tuples ...
2
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0answers
56 views

I have confirmed that the left integral diverges by the special value of the Lambert function in the equation $xe^{x^{2}}=1,$ anymore of it ??

Problem. Prove that $$\int_{0}^{\infty}\frac{{\rm d}x}{x^{2}e^{x^{2}}}= \infty$$ Indeed, that's true because $$\int_{0}^{\infty}xe^{x^{2}}{\rm d}x= 1/2\int_{0}^{\infty}{\left ( x^{2} \right )}'e^{x^{...
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1answer
49 views

$\Delta$ operator is it as a differential operator?

Consider the second law of dynamics written in terms of both momentum, momentum of a force and angular momentum, $$\bbox[5px,border:3px solid #F5B041]{\overline{F}=\frac{\Delta\overline{p}}{\Delta t}} ...
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2answers
73 views

Prove that $\sum_{n=1}^{\infty}n!x^n$ only converges if $x=0$

The question title basically sums up the goal of this question, that is, how to show that the power series $$\sum_{n=0}^{\infty}n!x^n $$ converges only if $x=0$. I will show my closest attempt at ...
0
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1answer
28 views

Convex set also has convex closure (alternative proof)

While studying convex analysis, i tried to write a proof for corollary 2 which i do not know if it is indeed correct. First statement(theorem): Let $S$ be a convex set in $R^{n}$ with a nonempty ...
2
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0answers
53 views

Given a recursion $a_{n+1}=\frac{a_n^2+1}{2}$ with $a_1=\frac12.$ Prove that $1-a_n\sim\frac2n-\frac{2\ln n}{n^2}\,{\rm as}\,n\rightarrow\infty$

Given a recursion $a_{n+ 1}= \dfrac{a_{n}^{2}+ 1}{2}$ with $a_{1}= \dfrac{1}{2}.$ Prove that $$1- a_{n}\sim\frac{2}{n}- \frac{2\ln n}{n^{2}}\,{\rm as}\,n\rightarrow\infty$$ Source: StachMath/@NN2 _ ...
2
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1answer
33 views

Alternative proof for $1+2w+3w^2+\ldots+nw^{n-1}= \frac{n}{w-1}$

I want to prove that if $w\neq 1$ is a root of unity, then $$F(w)=1+2w+3w^2+\ldots+nw^{n-1}= \frac{n}{w-1}$$ I have already proved it in two different ways but I am looking for a shorter and more ...
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1answer
69 views

Fermat's Last theorem alternative proof [closed]

Do the following statements lead to a simple proof of the theorem that $z = (x^n +y^n)^{1/n}$ is always irrational when $n\gt2$? When n is $\gt 1$ Then $(x + y) \gt z$, therefore a triangle can be ...
1
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1answer
47 views

Volume and orientation application -Calculus of manifolds

If $\omega$ is the volumen element of $V$ determined by $T$ and $\mu$, and $f\colon \mathbb{R}^n\to V$ is an isomorphism such that $f^*T=\langle,\rangle$, and such that $[f(e_1),\ldots,f(e_n)]=\mu$, ...
4
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1answer
137 views

Find the $\mathcal{O}\left(\frac{1}{n}\right)$ so that $a_n-1=\mathcal{O}\left(\frac{1}{n}\right)\,{\rm as}\,n\rightarrow\infty$

Given a recursion $a_{n+ 1}= \dfrac{a_{n}^{2}+ 1}{2}$ with $a_{1}= \dfrac{1}{2}.$ Find the $\mathcal{O}\left ( \dfrac{1}{n} \right )$ so that $$a_{n}- 1\sim\mathcal{O}\left ( \dfrac{1}{n} \right )\,{\...
1
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1answer
40 views

Basic proof that Isogeny = group homomorphism without Riemann Roch?

It is a well known fact that an isogeny $f:E_1\rightarrow E_2$ between elliptic curves (i.e., where $f(O)=O$) is a group homomorphism under the natural group structure of the elliptic curves. Every ...
5
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4answers
122 views

Given a recursion $a_{n+ 1}= \sqrt{na_{n}+ 2n+ 1}$ with $a_{1}\geq 1.$ Prove that $a_{n}\sim n\,{\rm as}\,n\rightarrow\infty$

Given a recursion $a_{n+ 1}= \sqrt{na_{n}+ 2n+ 1}$ with $a_{1}\geq 1.$ Prove that $$a_{n}\sim n\,{\rm as}\,n\rightarrow\infty$$ Let $b_{n}= \dfrac{a_{n}}{n},$ we need to prove $\lim b_{n}= 1,$ then ...
2
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1answer
95 views

Find the $\mathcal{o}\left(n\right)$ so that $a_n\sim\frac{\mathcal{o}\left(n\right)}{n}+\mathcal{O}\left(\frac{1}{n}\right){\rm as}\lim a_n=\infty$

Given a recursion $a_{n+ 1}= \dfrac{1}{2}\left ( a_{n}+ \dfrac{1}{a_{n}} \right )$ with $a_{1}> 0.$ Find the $\mathcal{o}\left ( n \right )$ so that $$a_{n}\sim\frac{\mathcal{o}\left ( n \right )}{...
3
votes
2answers
53 views

A simpler way to prove that $-x-\frac{1}{x+1}\leq-\frac{3}{2}$ for $x\geq1$, instead of computing derivative?

I have this function $$f(x)=-x-\frac{1}{x+1}$$ I want to prove that $f(x)\leq -\frac{3}{2}$ for $x\geq1$. We can easily prove this by calculating the first derivative which is negative, thus, the ...
0
votes
0answers
27 views

Gamma function identity proof

When trying to prove: $\Gamma\left(\dfrac{n}{2}\right)=\dfrac{2^{1-n}\sqrt{\pi }(n-1)!}{\left ( \frac{n-1}{2} \right )!}$, for all $n\in\mathbb{N}$, by mathematical induction, I cannot get a relation ...
3
votes
1answer
55 views

Question about smooth functions and their signs with given initial conditions

Problem Statement Suppose $f:\mathbb{R}\to\mathbb{R}$ is a smooth function (infinitely differentiable) where $f(x)\geq 0$ for all $x\in\mathbb{R}$, $f(0)=0$, and $f(1)=1$. Show that there is some $n\...
4
votes
1answer
125 views

Estimations of some new recurrence sequences

Problem 1. Given the recursion $a_{n+ 1}= \sqrt{a_{n}^{2}+ a_{n}}$ with $a_{1}= 1.$ Prove that $$a_{n}\sim\frac{n}{2}+ \frac{1}{2n}\,{\rm as}\,n\rightarrow\infty$$ For problem 1, I only can prove $a_{...
-2
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3answers
39 views

How would I prove that if 11n-5 is odd, then n is an even number using only a direct proof?

I've been able to prove this statement through contraposition and contradiction but I'm struggling to prove it through a direct proof. It seems I always get it in the form where 11n=2(k+3).
0
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0answers
22 views

$\sum_{k=m+2}^{\infty} a_k (w-w_0)^k = O(\lvert w-w_0\rvert^{m+2})$

Let $$\sum_{k=m+2}^{\infty} a_k (w-w_0)^k, a_k \in \mathbb{C}$$ be a convergent series on some open disk $B_r(w_0)$. I want to show that this series is of order $\lvert w-w_0\rvert^{m+2}$: $$\left\...
0
votes
3answers
82 views

Proving $\sum_{k=0} \frac{(-1)^k}{1+2k} = \frac{\pi}{4} $ taking $4 \int_0^1 \sqrt{1 - x^2} \; \text{d}x$ as the definition of $\pi$.

I'm trying to prove the Madhava-Leibniz formula for $\pi$ using one quarter of the area of the unit circle as the definition of $\frac{\pi}{4}$. $$ \int_{0}^{1} \sqrt{1-xx}\; \text{d}x = \sum_{k=0}^{\...
0
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0answers
7 views

Proof Check- Cartesian Product of countable sets.

I feel like my proof has holes somewhere. Like is there any explanation needed for why Ap x Ak+1 is countable?
6
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1answer
83 views

How to show two properties about the Cantor Set

Define $C_0=[0,1]$ and for $n\in\mathbb{N}$, define $$C_n=C_{n-1}\setminus\bigg(\bigcup_{k=0}^{3^{n-1}-1}\bigg(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\bigg)\bigg) $$ Then the Cantor set is defined as $$C=\...
0
votes
3answers
62 views

Proof that if: a^2=b^2 then a=b or a=-b

I have to prove that if a^2=b^2 then a=b or a=-b, and I came up with the following and I am wondering if it is valid use of absolute values?: a^2=b^2 |square root of both sides |a|=|b| |particularly ...
0
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0answers
11 views

PROOF: RSS is equal in the given models

I have a regular OLS model: Y = T + R where T_hat = ˆβ1 + ˆβ2Xi R_hat = ˆα1 + ˆα2Xi I am trying to prove that their RSS are equivalent. Is it the right approach? since RSS = Σ ˆu^2 = Σ(Yi -nˆβ1 - ...
5
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0answers
151 views

How to prove Craig's interpolation using amalgamation?

Notice: I have cross-posted this question into MathOverflow, here https://mathoverflow.net/questions/383999. I am looking for a sketch of, or a reference to, a proof which I can't seem to find in the ...
0
votes
1answer
36 views

A dumb question about $T_2$ space-Looking for alternative proof

Show that R is Hausdorff I have a proof for it I want to know is there anyway to do it using set theory alone For instance the distance between two sets , say U and V the formula is inf{||x-y||,x$\in ...
2
votes
2answers
65 views

If $b ∈ \Bbb Z$, then there is no integer a such that $b < a < b + 1$

If $b ∈ \Bbb Z$, then there is no integer $a$ such that $b < a < b + 1.$ I am doing an introductory course for mathematical proofs and this question is an exercise related to inequalities and ...
17
votes
2answers
958 views

How to find $\operatorname{P.V.}\int_0^1 \frac{1}{x (1-x)}\arctan \left(\frac{8 x^2-4 x^3+14 x-8}{2 x^4-3 x^3-11 x^2+16 x+16}\right) \textrm{d}x$?

The following problem is proposed by Cornel Valean:$$\operatorname{P.V.} \int_0^1 \frac{1}{x (1-x)}\arctan \left(\frac{8 x^2-4 x^3+14 x-8}{2 x^4-3 x^3-11 x^2+16 x+16}\right) \textrm{d}x=\log \left(\...

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