Questions on proving and manipulating inequalities.

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Proof By Induction $2^n \ge n^2$ for $n\ge4$

I am trying to prove the following, and here is what I have done: Can somebody help to complete this? $2^n \ge n^2$ for $n\ge 4$ $n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE Assume ...
2
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2answers
29 views

Proof By Induction $n^2 > 3n$ where $n\ge 4$

I am trying to prove the following example, however I seem to be getting a little stuck: For $n\in\mathbb N$, $n\ge 4, n^2>3n$ What I have Done: Base Case:$ n=4$, LHS: $4^2 = 16$, RHS: $3\cdot 4 ...
1
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1answer
61 views

Is $a \sin x + b \sin y \leq \sin(ax + by)$ true?

Studying math essay exam, I saw the following strange formula $$ a \sin x + b \sin y \leq \sin(ax + by), $$ where $x, y$ are arbitrary angles and $a + b = 1.$ Is the above inequality true, and can it ...
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1answer
17 views

Inequality regarding modulus

I am trying to prove this limit to be true: $$\lim_{x\to a}(x^2)=(a^2)$$ using the Epsilon Delta Limit Definition. So far I can understand how it works but I got stumped on this inequality ...
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1answer
48 views

Inequality with condition $x+y+z=xy+yz+zx$

I'm trying to prove the following inequality: For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that $$ \frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2} $$ My approach: After ...
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0answers
41 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
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4answers
403 views

Why is the Riemann sum less than the value of the integral?

Why is $ \frac{1}{n}\sum_{k=1}^n \frac{1}{1+\frac{k}{n}}\leq\int_0^1 \frac{dx}{1+x}=\log 2 $? Because I think: $$\int _0^1\frac{dx}{1+x}=\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}}$$ Why is the ...
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2answers
51 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
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4answers
41 views

How do we solve this inequality

I have the following inequation : $$\frac{1}{x-x^2-1}< 0$$ I know that the solution set will be all $x\in R$ but how do we find the answer? if we take the root of equation, we get imaginary ...
4
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3answers
96 views

Proving $\left(a+\frac{2}{a}\right)^2+\left(b+\frac{2}{b}\right)^2\ge \frac{81}{2}$ for all positive real $a,b$ such that $a+b=1$

I approached this problem in two different ways, but only one was successful. I'll post the latter as an answer, while here follows the first approach: I expanded the squares: ...
0
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1answer
33 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
0
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0answers
19 views

The $2$-norm of a Hermitian matrix does not exceed its $1$-norm

How to prove that the $2$-norm of a Hermitian matrix does not exceed its $1$-norm? In wiki, I see $2$-norm of matrix $A$ is $\le \sqrt{\|A\|_1\|A\|_\infty}$. But I don't know how to prove that ...
5
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5answers
87 views

Prove by mathematical induction: $\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1$

Could anybody help me by checking this solution and maybe giving me a cleaner one. Prove by mathematical induction: $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1; n\geq2$$. So after I check ...
2
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3answers
48 views

Inequalities and Differentiation

Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not ...
0
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1answer
30 views

inequality with gaussian cdf and density involved

in my calculations I've arrived at the following inequality $$ |\frac{4\phi(x)(1-2\Phi(x))}{(1+(1-2\Phi(x))^2)^2}| \leq 0.5 $$ where $\phi$ is Gaussian density, and $\Phi$ Gaussian cdf, which can ...
0
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4answers
68 views

Proof by Induction $3^n > n^3$

I am trying to prove the following, however I'm stuck at the Induction hypothesis Prove by induction that, for all integers $n$, if $n\geq 5$, then $3^n>n^3$ What I have Done: Base Case: $n ...
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2answers
35 views

Misunderstanding inequalities of integrals

We have to prove the following inequalities: 1) to show that $\frac{2x}{\pi }<sin\left(x\right)<x,\:and\:after\:1-e^{-\frac{\pi }{2}}\le \int _0^{\frac{\pi ...
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1answer
25 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
5
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1answer
125 views
+150

Prove a relationship involving floor functions

I am trying to prove that a particular expression is a lower bound for a very unusually-behaved function. The whole proof will be complete if I can just nail down the details of one technical lemma ...
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1answer
44 views

prove Inequalities for integrals

prove $\frac{\pi}{6}+\frac{1}{3}\leq \int_0^\frac{\pi}{2}\frac{1+\cos(x)}{2+\sin(x)}dx \leq \frac{\pi}{4}+\frac{1}{2}$ I got to the point where $\frac{1}{3} \leq f(x) \leq 1$, so $\frac{\pi}{6} ...
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3answers
54 views

Prove that $x^2 - 2013^2 \le y \le 2013^2 - x^2$ has an odd number of solutions

$x$ and $y$ are integers. $N$ is the number of solutions $(x, y)$ of this inequation $x^2 - 2013^2 \le y \le 2013^2 - x^2$. Prove that N is odd.
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3answers
77 views

Proving that $2^n+1\leq 3^n$ by induction

I need to prove the following using mathematical induction: $$2^n+1\leq 3^n\qquad\forall n\in\Bbb{Z^+}$$ Been working on this problem for a while and cannot figure it out. Any guidance or help would ...
2
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2answers
69 views

Prove $(\alpha_1 + … + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + … + \alpha_n^2)$

For any real numbers $\alpha_1, \alpha_2, . . . . ., \alpha_n$, $$(\alpha_1 + ...... + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ..... + \alpha_n^2)$$ And when is the inequality strict?
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0answers
33 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
0
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1answer
23 views

Satisfying a set of inequalities

Having a set of conditions (1)(2) and (3) as follows 1-$$\beta < 1$$ 2-$$ \beta > \alpha$$ 3-$$\alpha < 1$$ Can I say that the following inequality is incorrect? $$1-\alpha -\beta ...
7
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4answers
103 views

How to evaluate $\log x$ to high precision “by hand”

I want to prove $$\log 2<\frac{253}{365}.$$ This evaluates to $0.693147\ldots<0.693151\ldots$, so it checks out. (The source of this otherwise obscure numerical problem is in the verification ...
1
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3answers
68 views

Why $-1 \leq\frac{\langle A,B\rangle}{||A||\, ||B||}\leq1$?

I'm reading Apostol's Calculus. It says that due to the Cauchy-Schwarz inequality written as: $$|\langle A,B\rangle|\leq ||A||\, ||B||$$ Then $$-1\leq\frac{\langle A,B\rangle}{||A||\, ||B||}\leq ...
5
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2answers
64 views

Find multivariable limit $\frac{x^2y}{x^2+y^3}$

Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a ...
3
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1answer
47 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
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2answers
36 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...
1
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0answers
23 views

Simplification of inequalities

I am trying to simplify the following conditions $$\beta\leq1 $$ $$ 1-\alpha-\beta \leq 0$$ Can I say that the above two conditions are equivalent to saying that $$1- \beta \geq 0$$ $$\alpha \geq ...
1
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1answer
23 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
1
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1answer
28 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
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2answers
81 views

I can't understand how to prove this inequality

I don't understand how we can prove that inequality, without integration $$\frac{1}{x}\int_x^{2x} \left(2-\frac{1}{y+2}\right)\,dy \geq 2 - \frac{1}{x+2}.$$ P.S: Here is what I try... if can someone ...
2
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3answers
48 views

Inequality of factorial - Binomial coefficient

my name is Rafał and I decided to create this thread because of my inability to find a solution. I have been fighting with this inequality for 1.5 week and I have a hope that you will give me any hint ...
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1answer
33 views

Decide whether set is convex, connect and bounded.

Let $A=\{ \left(x,y,z \right)\in \mathbb{R}^3 : x^2+y^2-z^2+1<0\}$. Decide whether set A is: a) convex (definition i know: Set $A\in \mathbb{R}^k$ is convex set if for all $x,y \in A$ line segment ...
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1answer
48 views

Nobody can help me ? I can't believe that…

Okay, we have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}$, and we need to prove that: 1)$I_n\le log\left(2\right)$ $,\:\:\:\:\:$ why just log(2) ? can not be 1? ...
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2answers
69 views

Cauchy-Schwarz Hermitian Inner Product Remainder

A couple weeks ago, someone showed me a proof of Cauchy-Schwarz where he ended up deriving something of the form $$|\langle a,b\rangle|^2=|\langle a,a\rangle||\langle b,b\rangle| +f(a,b)$$ Where ...
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0answers
24 views

estimate an equality on sine function

I want to prove the following: For any given $\epsilon>0$, there exists a $\delta>0$ such that for any fixed $0<\theta<\frac{\pi}{2}$ with $\frac{\pi}{2}-\theta<\delta$, there exists ...
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1answer
37 views

Use the Mean Value Theorem to prove an inequality [on hold]

Using MVT, prove the following equation to be true: $$\sqrt{x} - \frac{x-y}{2\sqrt{y}} < \sqrt{y} < \sqrt{x} - \frac{x-y}{2\sqrt{x}},$$ given that $y>x>0$
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0answers
31 views

seeking for Newton's like inequalities as sufficient condition for polynomial to have only real zeros

For polynomial $P_n(x)=\sum_{k=0}^n a_k x^k, a_k>0$, it is known that a necessary condition for $P_n(x)$ to have only real zeros is that Newton's inequality holds: ...
2
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5answers
38 views

Complicated inequality, how to prove?

If $-\pi/2 <\theta<\pi/2$, how do I prove that $\left|\sin (\theta) - \sum_{k=0}^n (-1)^k \frac{\theta^{2k+1}}{(2k+1)!}\right| \leq \frac{|\theta|^{2n+2}}{(2n+2)!}.$
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2answers
51 views

If $x,y \in \mathbb{R}$ where $x\leq y$ and $y\leq x$. Does $x=y$?

I'm trying to complete this problem: Let $A$ be a nonempty set and suppose $\alpha$ and $\beta$ are both suprema of $A$. Prove that $\alpha = \beta$. The first thing i did was try to find an ...
2
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1answer
20 views

Complex modulus Inequality using $|exp(z)-1|$

I think I am almost there: Prove $\left|z\right|/4 < \left|\exp(z)-1\right|<7\left|z\right|/4$ for all $0<|z|<1$. MY ADVANCES First we note that $$ \left|\exp(z)-1\right| = ...
0
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1answer
32 views

How to apply Cauchy-Schwarz inequality to show an infinite series is bounded?

I would like to know why: $\left| \sum\limits_{t=0}^\infty \delta^tr(s_t,a_t)\right|\le \sum\limits_{t=0}^\infty \delta^t|r(s_t,a_t)|$ where: $\delta \in (0,1)$ and is a $r(s_t,a_t)$ is a real ...
12
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0answers
104 views
+50

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discuss olympiad problem none of student solve it,therefore, meaning this problem is so hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
4
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1answer
71 views

Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function such that $\int\limits_0^1f(x)dx=0$. Prove that $$\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$$ My approach as follow Let ...
1
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0answers
30 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
0
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0answers
34 views

Prove a lower bound for a/b+b/c+c/a

Given $a,b,c>0$, prove the inequality $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq \frac{(1+a^2)}{(1+a\dot\,b)} + \frac{(1+b^2)}{(1+b\dot\,c)} + \frac{(1+c^2)}{(1+c\dot \,a)}$$ I have tried ...
0
votes
0answers
8 views

interval boundaries for covariance of two variables, given their covariance with a third variable

I am thinking about three random variables $X, Y, Z$. If the correlation coefficient (alternatively, covariance, assuming $Var(X,Y,Z)=1$) between variables $X,Y$ is given, as is correlation between ...