Inequality Definition and 1000 Threads

Hello! (Wave)Given that $n \geq 15$, how can we conclude the following? (Thinking) $$cn \lg n- cn \lg \left ( \frac{3}{2}\right )+\frac{n}{2}+15 c \lg n-15 c \lg \left ( \frac{3}{2}\right) \leq cn \lg n, \text{ for } c>1 \text{ and } n>15$$

Prove that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$.

Homework Statement its the second one. let n∈ℕ \ 0 and k∈ℕ show that (n choose k) 1/n^k <= 1/k! Homework Equations axioms of ordered fields? The Attempt at a Solution [/B]i have been working on this all afternoon. I know 0<k<n since its a requirement for (n choose k). I've tried...

Not sure if this was the right place for this but here goes. Hello all, so I'm trying to get an intuitive grasp of the Clausius Clapeyron relation dP/dT= L/TdelV. Where L is the latent heat of the phase transition. What I've got so far is this; the relation tells you how much extra pressure must...

Homework Statement Dear Mentors and Helpers, Here's the question: Find the possible values of k such that one root of the equation 2x^2 + kx + 9 = 0 is twice the other. Homework Equations My classmate's working: Discriminate > 0 k^2 - (4)(2)(9) > 0 k^2 -72 > 0 [k + sqrt (72)] [k- sqrt(72)] >...

Prove that for $r>2$ we have $$\frac{\zeta\left(r\right)}{\zeta\left(2r\right)}<\left(1+\frac{1}{2^{r}}\right)\frac{\left(1+3^{r}\right)^{2}}{1+3^{2r}}.$$ I've tried to write Zeta as Euler product but I haven't solve it.

I'm working on this problem. $$2x^2 + 4x \ge x^2 - x - 6$$ I got here $$2x -x \ge -3$$ But I don't know how to go from here.

Need help with this exercise been stuck on it for a while i think i get the gist of what i am supposed to do but can't seem to get it to work i am definitely missing something. I set h(x)= x-2/3 - g(x) and tried using the mean value theorem on [a,b] and then tried finding the minimum value of...

Which of the following have the least value if $$-1 < x < 0$$ $$(A) -x$$ $$(B) 1/x$$ $$(C) -1/x$$ $$(D) 1/x^2 $$ $$(E) 1/x^3$$ Mmmmmmmm... I'm not sure what to do, but I'll definitely try. We can break it up into two inequalities. $$ x > -1$$ $$0 > x$$ $$\implies -x < 1, 0 < -x$$...

I've been reading "The Qualitative Theory of Ordinary Differential Equations, An Introduction" and am now stuck on an inequality I am supposed to be able to prove. I am pretty sure the inequality comes from linear algebra, I remember seeing something about it in my intro class but I let a friend...

Bell's 1971 derivation The following is based on page 37 of Bell's Speakable and Unspeakable (Bell, 1971), the main change being to use the symbol ‘E’ instead of ‘P’ for the expected value of the quantum correlation. This avoids any implication that the quantum correlation is itself a...

This is problem 20b from chapter I 4.10 of Apostol's Calculus I. The geometric mean G of n positive real numbers x_1,\ldots, x_n is defined by the formula G=(x_1x_2\ldots x_n)^{1/n}. Let p and q be integers, q<0<p. From part (a) deduce that M_q<G<M_p when x_1,x_2,\ldots, x_n are not all...

Hello, I have this inequality: $$-x^2 + 4 < 0$$ Then, I get to $$-(x-2)(x+2) < 0$$ Now, how do I solve this question from here. I understand that x = -2, or x =2 but how do I use this to solve the inequality? Thanks

Solve the Ineqality $$x^2 + 2x -8 \le 0$$I know enough to factor it like this $$(x-4) (x+2) \le 0$$ So I get 4 and -2. I just don't know how to get to the answer from here which is: $$x \ge -4\cup x\le 2$$ unless I'm misreading the answer incorrectly. Thanks

Homework Statement Use M.I. to prove that n! > n^3 for n > 5 The Attempt at a Solution I already proved n! > n^2 for n>4, but this is nothing like that. This is my inductive step so far. n=k+1 (k+1)! > (k+1)^3 (k+1)! - (k+1)^3 > 0[/B] (k+1)! - (k+1)^3 = (k+1)[k! - (k+1)^2]...

Bell, QM Ideas - Science 177 1972 :" Strictly, however. a hidden variable theory could be non-deterministic; the hidden variable could evolve randomly (possibly even discontinuously) so that their values at one instant do not specify their values at the next instant" From the locality...

http://en.wikipedia.org/wiki/CHSH_inequality#Bell.27s_1971_derivation The last step of the CHSH inequality derivation is to apply the triangle inequality. I see there are relative polarization angles, but I don't see any sides have defined length to make up a triangle. Where is the triangle?

Hi MHB, When I first saw the problem (Prove that $\sin^2 x<\sin x^2$ for $0\le x\le \sqrt{\dfrac{\pi}{2}}$), I could tell that is one very good problem, but, a good problem usually indicates it is also a very difficult problem and after a few trials using calculus + trigonometry method, I...

There are real numbers $l,\,m,\,n$ such that $l\ge m\ge n >0$. Prove that $\dfrac{l^2-m^2}{n}+\dfrac{n^2-m^2}{l}+\dfrac{l^2-n^2}{m}\ge 3l-4m+n$.

If $a,b\in \mathbb{R}^{+}.$ Show that $a>b\implies a^{-1}<b^{-1}.$

I have literally spent all day reading and am still very much in the dark. First off does anyone have a link to detailed blow by blow account that doesn't assume an understanding of advanced maths and physics concepts and notations but will actually address the issue in depth? Here are...

Here's my first challenge! Let $f : [0,1] \to \Bbb R$ be continuously differentiable. Show that $\displaystyle \left|f\left(\frac{1}{2}\right)\right| \le \int_0^1 |f(t)|\, dt + \frac{1}{2}\int_0^1 |f'(t)|\, dt$.

Homework Statement Find all integer roots that satisfy (3x + 1)/(x - 4) < 1. The Attempt at a Solution I would do this: Make it an equation and find x such that (3x + 1)/(x - 4) = 1. 3x + 1 = x - 4 2x = -5 x = -5/2 Then check if the inequality is valid for values smaller than x and for...

hi there, I am trying to prove the following inequality: let z\in \mathbb{D} then \left| \frac{z}{\lambda} +1-\frac{1}{\lambda}\right|<1 if and only if \lambda\geq1. The direction if \lambda>1 is pretty easy, but I am wondering about the other direction. Thanks in advance

Hi everyone! First of all thank you for a great forum! I downloaded the app and find it ingenious! The problem stated above is from "3000 Solved Problems in Calculus". The book solves this problem simply by stating: "No. Let a=1 and b=-2". However, I am curious to know if it is possible...

This is from section I 4.9 of Apostol's Calculus Volume 1. The book states the Cauchy-Schwarz inequality as follows: \left(\sum_{k=1}^na_kb_k\right)^2\leq\left(\sum_{k=1}^na_k^2\right)\left(\sum_{k=1}^nb_k^2\right) Then it asks you to show that equality holds in the above if and only if there...

Hi guys, I have to teach inequality proofs and am looking for an opinion on something. Lets say I have to prove that a2+b2≥2ab. (a very simple example, but I just want to demonstrate the logic behind the proof that I am questioning) Now the correct response would be to start with the...

Homework Statement Let b and a be two complex numbers. Prove that |1+ab| + |a + b| ≥ √(|a²-1||b²-1|). Homework Equations Complex algebra The Attempt at a Solution I don't know how to proceed. I posted it here to get some ideas :p

given:$a>b>c>0$ prove:$a^{2a}b^{2b}c^{2c}>a^{b+c}b^{c+a}c^{a+b}$

Let $x\ge \dfrac{1}{2}$ be a real number and $n$ a positive integer. Prove that $x^{2n}\ge (x-1)^{2n}+(2x-1)^n$.

Definition/Summary the Schwarz inequality (also called Cauchy–Schwarz inequality and Cauchy inequality) has many applications in mathematics and physics. For vectors a,b in an inner product space over \mathbb C: \|a\|\|b\| \geq |(a,b)| For two complex numbers a,b : |a|^2|b|^2...

Definition/Summary Arithmetic-Geometric-Harmonic Means Inequality is a relationship between the size of the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) of a given set of positive real numbers. It is often very useful in analysis. Equations {\rm AM} \geq {\rm GM}...

Can anyone help me prove the greatest integer function inequality- n≤ x <n+1 for some x belongs to R and n is a unique integer this is how I tried to prove it- consider a set S of Real numbers which is bounded below say min(S)=inf(S)=n so n≤x by the property x<inf(S) + h we have...

prove : $\dfrac {1\times 3 \times 5\times------\times 99}{2\times 4\times 6\times --\times {100}}>\dfrac {1}{10\sqrt 2}$

prove : $18<1+\dfrac {1}{\sqrt 2}+\dfrac {1}{\sqrt 3}+----+\dfrac{1}{\sqrt {99}}<19$

Hey! (Mmm) I have to define an asymptotic upper and lower bound of the recursive relation $T(n)=3 T(\frac{n}{3}+5)+\frac{n}{2}$. Firstly,I solved the recursive relation: $T'(n)=3 T'(\frac{n}{3})+\frac{n}{2}$,using the master theorem: $$a=3 \leq 1, b=3>1, f(n)=\frac{n}{2} \text{ asymptotically...

Convergence of Divergent Series Whose Sequence Has a Limit Homework Statement Suppose ∑a_{n} is a series with lim a_{n} = L ≠ 0. Obviously this diverges since L ≠ 0. Suppose we make the new series, ∑(a_{n} - L). My question is this: is there some sufficient condition we could put solely...

This is problem 13 from section I 4.7 of Apostol's Calculus Volume 1: Prove that 2(\sqrt{n+1}-\sqrt{n})<\frac{1}{\sqrt{n}}<2(\sqrt{n}-\sqrt{n-1}) if n\geq 1. Then use this to prove that 2\sqrt{m}-2<\displaystyle\sum_{n=1}^m\frac{1}{\sqrt{n}}<2\sqrt{m}-1 if m\geq 2. I have proved the first...

Prove that $\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}\ge10x-x^2$.

I thought it might be interesting to point out this article: Title: Violation of Bell's inequality in fluid mechanics Authors: Robert Brady and Ross Anderson (Cambridge) Abstract:

What are some simplified conditions for which: $$W\bigg(A-\frac{X}{W}\bigg)^3\bigg[X-AW-\frac{AY}{N}(B+D)-\frac{AZ}{N}(C+D+E+F+G)\bigg]+\frac{X}{N}\bigg[Y(A+H)(B+D)+AZ(C+D+E+F+G)\bigg]<0$$ **WHERE:** All of the letters are positive parameters (not constants) and: $1.$ $$A,B,C,D,E,F,G,H < N...

Hello, I do not know if this is the right place to post this question, but I believe it falls under algebra. Please redirect me if appropriate. Question: How can I show that $$P-QR^3<\frac{R^4}{C}$$ for $$C,P,Q,R > 0?$$ Thanks.

$0<\alpha, \beta<\dfrac {\pi}{2}$ prove :$\dfrac {1}{cos^2\alpha}+\dfrac {1}{sin^2\alpha\,sin^2\beta\, cos^2\beta}\geq 9$ and corresponding $\alpha$, and $\beta$

Prove that $6^{33}>3^{33}+4^{33}+5^{33}$.

Homework Statement This isn't really a problem so much as me not being able to see how a proof has proceeded. I've only just today learned about Dirac notation so I'm not too good at actually working with it. The proof in the book is: |Z> = |V> - <W|V>/|W|^2|W> <Z|Z> = <V - ( <W|V>/|W|^2 ) W|...

Given that $0<k,\,l,\,m,\,n<1$ and $klmn=(1-k)(1-l)(1-m)(1-n)$, show that $(k+l+m+n)-(k+m)(l+n)\ge1$.

I am trying to find the minimum of the following expression: $$\frac{\sin^2x+8\cos^2x+8\cos x+\sin x}{\sin x\cos x}\,\,\,,0<x<\frac{\pi}{2}$$ I know I can bash this with calculus but the expression has a nice minimum value (=17) which makes me think that it can be solved by use of some...

I never heard of the CSHS Inequality until I read it in another thread. The other interesting item was this: I think an important part of that discussion is the more hits are ignored, the easier it is for local realistic theories to score over 2.00. So I just had to try. For those familiar...

The problem statement Let ##A ∈ K^{m×n}## and ##B ∈ K^{n×r}## Prove that min##\{rg(A),rg(B)\}≥rg(AB)≥rg(A)+rg(B)−n## My attempt at a solution (1) ##AB=(AB_1|...|AB_j|...|AB_r)## (##B_j## is the ##j-th## column of ##B##), I don't know if the following statement is correct: the columns of...