Questions on proving, manipulating and applying inequalities.

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Proving the convergnce of a sequence

So, I have to prove that the sequence defined as $a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit. I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. ...
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31 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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0answers
40 views

Proving $n^n \cdot \left(\frac{n+1}{2}\right)^{2n}\geq \left(\frac{n+1}{2}\right)^3$

Proving $$n^n \cdot \left(\frac{n+1}{2}\right)^{2n}\geq \left(\frac{n+1}{2}\right)^3$$ , Where $n\in \mathbb{N}$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}\;,$ We get ...
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4answers
57 views

How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove the following inequality- $$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without the help of induction? Thanks for any help!!
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3answers
56 views

Prove that $\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without using induction? Note that clearly $n\neq 0$ Thanks for any help!!
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5answers
74 views

Prove that $\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$

Without using Mathematical Induction, prove that $$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$ I am unable to solve this problem and don't know where to start. Please help me ...
3
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1answer
50 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
3
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1answer
25 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
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Understanding ADMM: how is it applied to this particular problem?

While reading Boyd's paper on ADMM I encountered an issue. Consider the following problem: Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and ...
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2answers
28 views

Triangle Inequality?

I'm having trouble proving the following claim: $\forall a, b, c \in \mathbb{R}_+: T(a, b, c) \Rightarrow [|a − b| < c$ and $|b − c| < a$ and $|a − c| < b]$ Where $T(a, b, c)$ is a ...
2
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1answer
26 views

Prove that $|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$

Prove for any $a, b \in [-1, 1]$ that $$|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$$ I'm sure there is a solution using the Cauchy-Swartz inequality. Thus i tried to ...
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0answers
22 views

Prove Jensen Inequality holds for a function

Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$ Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, ...
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0answers
15 views

How to prove the following equivalence

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ I want to prove that $\liminf_k ...
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4answers
69 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
2
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3answers
70 views

Trying to prove for all integers: $n \ge 1 \implies \frac{2n+1}{2n+2} \ge \frac{\sqrt{n}}{\sqrt{n+1}}$

Been racking my brain on this one.. I've tried some things but not sure if it flows logically: $\forall x \in \mathbb{Z}: n \ge 1$ $n+2 \ge 1$ $2n+2 \ge n+1$ $\frac{2n+1}{2n+2} \ge ...
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2answers
64 views

how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$

Prove that if $a,b,c$ are non-negative real numbers, then $(ab + bc + ca)^2 \geq 3abc(a+b+c)$. I tried to compute from $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$.
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2answers
42 views

Is this function bounded above?

Consider nonconstant functions $f(x), g(x) \neq x$. Suppose there exist positive constants $k_1$ and $k_2$ such that $k_{1} x \leq f(x) \leq k_{2} x$ and $\frac{1}{2}k_{1} x \leq g(x) \leq k_{2} x$. ...
3
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2answers
120 views

Prove that $7<e^2<8$

I was asked by my teacher to prove that $7<e^2<8$ using only algebraic methods and knowing that $2<e<3$. I don't know how to do this, where to start from, but I guess that I would need ...
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0answers
9 views

What function gives this inequality?

Let $i<j<k<l$ be positive integers. I want to find a "nice" function $f(x, y)$ such that $f(i, k)+f(j, l)>\max(f(i, j)+f(k, l), f(i, l)+f(j, k))$. This seems a bit tricky because the ...
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2answers
25 views

Question on double inequality with radicals

A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$? Thank you very much.
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1answer
78 views

What is the intuition behind the Cauchy-Schwarz inequality in the real numbers?

The Cauchy-Schwarz inequality states that $$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$ The proof, with the discriminant argument, is ...
3
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2answers
46 views

Sketch the set of points satysfing an inequality $|z+1|+|z-1|\leq2$

The inequality is $$|z-1|+|z+1|\leq2$$ I used a triangle inequality to show that Since triangle inequality states: $$|z+w|\leq|z|+|w|$$ Then $$|z-1+z+1|\leq|z-1|+|z+1|\leq2$$ So $$|2z|\leq2$$ From ...
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0answers
9 views

Inequality problem for Markov Process

Is there any upper bound available for the following quantity $$E[\max_{1 \leq k \leq n} X_k]$$ where $\{X_n\}$ is a Markov chain.
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1answer
28 views

Question on statement of Cauchy-Schwarz inequality: $\vert\langle x,y \rangle \vert \leq \Vert x \Vert \cdot \Vert y \Vert$

Denoting the Cauchy-Schwarz inequality as Wikipedia does, $$\vert\langle x,y \rangle \vert \leq \Vert x \Vert \cdot \Vert y \Vert$$ and noting that $$\vert\langle x,y \rangle \vert = \Vert x\cdot y ...
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2answers
57 views

An inequality on an arbitrary function

I'm trying to find the complexity of a program and reduced the question to the following one: Let $g$ be a function from natural numbers (including $0$) to natural numbers. Assume that for every $n ...
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1answer
22 views

Solving inequality to state function f>1

I have the following function, $$f = \frac{x-a}{y-a} $$ I want to specify the condition for $f>1$ I wrote it as, $$f>1$$ when $$\frac{x-a}{y-a}>1$$ so, I rewrote it as, $$x-a>y-a$$ ...
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3answers
41 views

Why are some solutions excluded if we simply multiply the denominator in an fraction inequality?

I have the next inequality for which I have to find the solutions : $$\frac{2x-5}{3x-1}\geq 1, \text{ where } x\in\mathbb{R}$$ I know I have to subtract $1$ and then I have to analyse the sign for ...
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1answer
29 views

How to solve the “$\min$” function coming with absolute value in an inequality

Prove that if $|x-a|<\min(k/(2|1+b|),1)$ and $|y-b|<(k/2(1+|a|))$, then $|xy-ab|<k.$ In this question I don't know how to deal with "$\min$" part in the first inequality.
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1answer
26 views

How to show that $(X-a)^+\le X^++|a|$

How to show that $(X-a)^+\le X^++|a|$, where $X, a$ are real Is the following OK; $(X-a)^+ +(a-X)^+=|X-a|$ and If the claim (in the yellow box) is not true then also; $(a-X)^+> a^++|X|$ but ...
3
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2answers
43 views

Proof of $a^x ≥ x+1 \; \forall x \in \Bbb R \implies a=e$

I'm trying to prove the following : Let $a>0$ a real number. Then : $\quad a^x ≥ x+1 \;\; \forall x \in \Bbb R \iff a=e$ I managed to prove the '$\Longleftarrow$' part : $x≥0$ then ...
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3answers
42 views

Solving inequlity with $e^x$

I'm studying differential calculus, but one of the questions involves solving an inequality: $$(x-2)e^x < 0$$ I intend to go deeper in solving inequalities later, but I just want to understand ...
3
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1answer
111 views

Inequality $2a^nb^nc^n+1\geq a^{2n}+b^{2n}+c^{2n}$

Let $a,b,c\in[-1,1]$ be such that $$2abc+1\geq a^2+b^2+c^2.$$ Prove that $$2a^nb^nc^n+1\geq a^{2n}+b^{2n}+c^{2n}$$ for any positive integer $n$. The case $n=1$ is of course the same as the ...
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0answers
27 views

A little help with inequalities (homework) . [on hold]

Can anyone please verify my answers for this problem? $a,b,c,d$ are rational numbers. Answers should be true or false. If $a<b$ and $c<d$ then $\frac{a}{c} < \frac{b}{d}$ to which my answer ...
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1answer
36 views

Inequality for proof of Density Theorem

Someone could help me white this question or indicate some reference? Lemma:For any $\epsilon>0$ There exits a $C=C(n,\epsilon)$ such that for $u \in H^{1}(B_{1})$ with $|\{ x \in B_{1} ; ...
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3answers
54 views

Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
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0answers
30 views

Use the arithmetic-geometric inequality for this list to deduce the arithmetic-geometric inequality for $n$.

Suppose that $n$ is not a power of two. Let $2^k$ be a power of $2$ that exceeds $n$ and consider the list $$a_1,\dots,a_n,\underbrace{A,A,\dots,A}_\text{$2^k-n$ times}$$ of length $2^k$. Use the ...
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1answer
26 views

How to find the real values of the parameter a, so that the inequality doesn't have positive solutions?

The inequality is the following: $$ \frac{x^2 - 6ax + 2x - 5a - 1}{x+a+1} < 0 $$ If we write this inequality like $\frac{a}{b} < 0$, I may say that for $a=0$, I found that $X_1=3a - 1 - ...
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2answers
88 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
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1answer
19 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
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1answer
57 views

Prove inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$

Prove the inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ For all $n \in \mathbb{N}$ I've done the right hand side, but can't do the left side of the inequality. For ...
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2answers
38 views

the minimum value of $a^2+b^2-a-\frac{2b}{3}$ using known standard inequalities

If $a,b$ are real numbers satisfying $a+2b=3,$ then the minimum value of $a^2+b^2-a-\frac{2b}{3}$ Here $a+2b=3\implies a=3-2b$ $a^2+b^2-a-\frac{2b}{3}=(3-2b)^2+b^2-(3-2b)-\frac{2b}{3}$ ...
3
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2answers
48 views

Prove $ab\leq F(a)+G(b),~~\text{for all}~a\geq 0$ and $b\geq 0$

Suppose that te function $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous and strictly increasing, with $f(0)=0$ and $f([0,\infty))=[0,\infty)$. Then define ...
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0answers
11 views

Around Shapiro's inequality

Shapiro´s inequality is announced as follows: For $n$ even with $0\lt n\le 12$ and for $n$ odd with $0\lt n\le 23$ one has ...
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1answer
22 views

Bounds on sum of entries of an idempotent symmetric matrix

Suppose that $M$ is symmetric and idempotent, dimensions $n\times n$, and trace $n-k$. Let $e$ ($n\times 1$) be a column of $1$'s. Let $$ S_1\equiv e'Me,\quad ...
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0answers
23 views

How to solve inequalities where the $x$ term appears inside the argument of multiple different functions?

We're asked to study the sign of the following function: $$\frac{x(\ln{x}+1)^2 - 2(\ln{x}+1)^2 - \frac{4}{x(\ln{x}+1)}}{(x(\ln{x}-1)^2)^2} \geq0,$$ in which the $x$ variable appears both outside ...
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4answers
55 views

Prove the following Inequality [3]

Suppose $m>x$, where $x$ belongs to positive Real Numbers and letting $n\ge m$, show that $${x^n\over n!}\le{x^n\over m^n}\cdot{m^m\over (m-1)!}$$ Frankly, I consider myself as a beginner in ...
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0answers
19 views

Comparing irrational powers

I want to compare two numbers of the form $a^b$ and $b^a$ for $a,b, >1$. The simpler way that I know is to start from: $$ a^b\le b^a \quad \iff \quad b\log a\le a \log b \quad \iff \quad ...
1
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1answer
55 views

Prove that $(e^x-\ln y)^2+(x-y)^2\geq 2$

Prove that $(e^x-\ln y)^2+(x-y)^2\geq 2,\forall x\in\mathbb{R},\forall y>0$. When does the equality hold? I note that the equality holds only if $x=0,y=1$, but I don't know how to prove that ...
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0answers
31 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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1answer
41 views

Not understanding a cancellation step in an inequality proof from Spivak's Calculus.

have just been reading through and doing the questions in Spivak's Calculus, but am not entirely sure I am understanding a step in one of the proofs given in chapter $5$ (but was given as an exercise ...