Inequality Definition and 1000 Threads

Homework Statement Show that if 0 \le x \le a, and n is a natural number, then 1+\frac{x}{1!}+\frac{x^2}{2!}+...+\frac{x^n}{n!} \le e^x \le 1+\frac{x}{1!}+\frac{x^2}{2!}+...+\frac{x^n}{n!}+\frac{e^ax^{n+1}}{(n+1)!} Homework Equations I used Taylor's theorem to prove e^x is equal to the LHS...

For any triangle $ABC$, prove that $\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$

Problem: Prove: $$\sqrt{C_1}+\sqrt{C_2}+\sqrt{C_3}+...+\sqrt{C_n} \leq 2^{n-1}+\frac{n-1}{2}$$ where $C_0,C_1,C_2,...,C_n$ are combinatorial coefficients in the expansion of $(1+x)^n$, $n \in \mathbb{N}$. Attempt: I thought of using the RMS-AM inequality and got...

Let $x, y$ be real numbers such that $9x^2+8xy+7y^2 \le 6$. Show that $7x+12xy+5y \le 9$.

Homework Statement Let 0 ≤ x_1 ≤ x_2 ≤ ... ≤ x_n and x_1 + x_2 + ... + x_n = 1 , All the 'x' are real and n is a natural number. Prove the following: (1+x_1^21^2)(1+x_2^22^2)...(1 + x_n^2n^2) ≥ \frac{2n^2+9n+1}{6n} Homework Equations The Attempt at a Solution...

I'm stuck on one aspect of the proof on page 105 of the 2nd edition. Equation 6.13 is necessary for the inequality to be an equality as it says but they never seem to account for inequality 6.11. Specifically, I don't see how this satisfies 2 Re<u,v> = 2 |<u,v>| Thanks for any guidance.

Here's a claim: Assume that a function f:[a,b]\to\mathbb{R} is differentiable at all points in its domain. Then the inequality |f(b) - f(a)| \leq \int\limits_{[a,b]}|f'(x)|dm(x) holds. The integral is the Lebesgue integral. Looks simple, but I don't know if this is true. There exists...

Are there any modern interpretations of QM that predict the correlations in a Bell Inequality violation ? Preferably a local non realistic model based on mechanisms.

Homework Statement I am asked to discover and prove the inequality relationship between root mean square, arithmetic mean, geometric mean, and harmonic mean. Homework Equations Let a,b, be non-negative integers. (a-b)2 ≥ 0 and (√a-√b)2 ≥ 0 The Attempt at a Solution Using (a-b)2 ≥...

Leaving aside the debatable point about whether bell's violation rules out all local theories or just local realism, is there general agreement that if the assumptions are valid, violation of Leggett's inequalities rules out any non-local model that treats properties other than position as real...

[solved]Complex inequality Homework Statement You have the two inequalities, where k is a complex number; |k+\sqrt{k^2-1}|<1 and |k-\sqrt{k^2-1}| <1 Show that if ##|k|>1##, then the second inequality is fulfilled, while the first one is impossible for any value of k. The Attempt...

Homework Statement Question: Use induction to prove that 3^n > n x 2^n for every natural number n ≥ 3 Homework Equations N/A The Attempt at a Solution Answer: Step 1: 3^3 > 3 x 2^3 ⇒ 27 > 24 Step 2: Assume 3^k > k x 2^k Step 3: 3^(k+1) > (k+1) x 2^(k+1) ⇒ 3 x 3^k > k x 2^(k+1) +...

∫dQ/T≤∫dQ(rev)/T * , where both integrals are evaluated between the same thermodynamic coordinates- A and B , say. - I am having trouble interpreting this inequality. -( I understand the derivation in my textbook via the Clausius diagram(considering a reversible and an ireversible process...

Homework Statement Let f(x)=1-x-x3. Find all the real values of x satisfying the inequality, 1-f(x)-f3(x)>f(1-5x).Homework Equations The Attempt at a Solution I honestly don't know how to start with this one. Substituting f(x) directly in the inequality doesn't look like a good idea. I need a...

Homework Statement Solve for k: k2 - 16k < 0 In the answer it has 0 < k < 16, I do not know how they get there from the original question.

Solve 2x - |x+1| < 4.

Question http://puu.sh/52zAa.png Attempt http://puu.sh/52AVq.png I've attempted to use Riemann sums and use the integral to prove the inequality, not sure if this was the right approach to start with as I am now stuck and don't see what to do next. For part (b), I know that if (2√n...

Homework Statement Prove the inequality double integral (dA / (4+x^2+y^2)) is less than or equal to pi, where the double integral has a sub D where D is the disk x^2 + y^2 less than or equal to four Homework Equations The Attempt at a Solution I really have no idea, anyone want to...

Homework Statement |x + y| ≥ |x| - |y| [Hint: write out x = x + y - y, and apply Theorem 3, together with the fact that |-y| = |y|] Homework Equations Theorem 3: |a + b| ≤ |a| + |b| x = x + y - y |-y| = |y| The Attempt at a Solution |x + y| ≥ |x| - |y| x = x + y - y (don't know where to...

(I wasn't sure how to title this, it's just that the statement resembles Chebychev's but with two RV's.) Homework Statement Let \sigma_1^1 = \sigma_2^2 = \sigma^2 be the common variance of X_1 and X_2 and let [roh] (can't find the encoding for roh) be the correlation coefficient of X_1 and X_2...

Could I get some help with this question? Please refer to the attachment. Thanks

I have been reading about the derivation of Clausius' Inequality and there are a few things I do not understand. I have attached an image of the cycles. B) shows one carnot engine performing work ##d W_i## per cycle and delivering heat ##d Q_i## per cycle. For ##T'## to remain unchanged, it...

Hi, I'm trying get a better understanding of Bell's inequality in the form $$\left|E\left(\bf{a},\bf{b}\right) -E\left(\bf{a},\bf{c}\right)\right|\leq 1+E\left(\bf{b},\bf{c}\right)\enspace.$$ I'm considering the Bell state $$\left|\psi\right\rangle=...

I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just...

I am given a statement to prove: Show (without using the Binomial Theorem) that \((1+x)^n\geq{1+nx}\) for every real number \(x>-1\) and natural numbers \(n\geq{2}\). I am given a hint to fix \(x\) and apply induction on \(n\). I started by supposing \(x\) is a fixed, real number larger than -1...

The following inequality can easily be proved on ##ℝ## : ## ||x|-|y|| \leq |x-y| ## I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using...

Prove that for any two numbers x,y we have [(x^2 + y^2)/2] >= x + y - 1 Solution) For any number a we have have a^2 > 0. So, (x-1)^2 + (y-1)^2 >= 0 And if we solve this we get the solution.I don't get the red part.

Using only the axioms of arithmetic and order, show that: for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y I'm really lost on where to start, my attempt so far was this as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the...

## x - |x-|x|| > 2 ## how would I go about solving something like this? my initial thoughts was to consider if x >= 0 I get 2-x < 0 then x > 2 in that case then consider if x < 0 which I get -|x+x| > 2-x then 2x > 2-x then x > 2/3 but I'm having troubles deciding which one is correct, and if...

Hi guys, Can you help me I am stuck: By finding the real and imaginary parts of z prove that, $$|\sinh(y)|\le|\sin(z)|\le|\cosh(y)|$$ i have tried the following: Let $$z=x+iy$$, then $$\sin(z)=sin(x+iy)=\sin(x)\cosh(y)+i\sinh(y)\cos(x)$$ $$|\sin(z)|=\sqrt{(\sin(x)\cosh(y))^2+(\sinh(y)...

Hello. I am reading an introduction to induction example, and I am having the hardest time trying to determine what exactly happened in the proof. Can somebody please help? How can ##3^{k-1}## + ##3^{k-2}## + ##3^{k-3}## all of a sudden become ##3^{k-1}##+##3^{k-1}##+##3^{k-1}## and how can be...

here is the inequality: ##(\sum\limits_{i=1}^n |x_i-y_i|)^2= \ge \sum\limits_{i=1}^n(x_i-y_i)^2+2\sum\limits_{i \neq j}^n |x_i-y_i|\cdot |x_j-y_j|## does it have a name/is the consequence of a theorem? Thank you :)

Suppose that ##F(u,v)=a|u+v|^{r+1}+2b|uv|^{\frac{r+1}{2}}##, where ##a>1, b>0## and ##r\geq3.## How we can show that there exists a positive constant c such that ## F(u,v)\geq c\Big( |u|^{r+1}+|v|^{r+1}\Big). ##

Homework Statement What are the possible values of |2x−3| when 0<|x−1|<2? Homework Equations The Attempt at a Solution We know \left|x-1\right| becomes x-1 if x-1≥0 and -(x-1) if x-1<0. Now consider two cases. Case 1: 0<x-1<2 \Rightarrow 1<x<3 \Rightarrow -1<2x-3<3. Case...

Log x ((x+3)/(x-1) > Log x x ?? I've managed to find 4 conditions for this inequality: 1. -1 > x > 3 2. x > -3 3. x > 0 4. x ≠ 1 but I'm not sure how to write the solution. Is it " 0 < x & 1 < 0 < 3 " ? Thanks.

Homework Statement Prove; n^2 > n+1 for n = 2,3,4 by Induction Homework Equations The Attempt at a Solution p(n)= P(2) 2^2> 2+1 --> 4>3 Induction step: P(n+1): (n+1)^2 > (n+1) +1 (n+1)^2> n+2 n^2 + 2n + 1 > n+2 | -n n^2 +n + 1> 2 | -1 n^2 +n > 1 Is this correct, and how...

Hello all, I am currently reading about the triangle inequality, from this article http://people.sju.edu/~pklingsb/cs.triang.pdf I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the...

Prove x^x \ge \left( \frac{x+1}{2} \right)^{x+1} for $x>0$.

This form of a Bell inequality: n[x-y+] + n[y-z-] ≥ n[x+z+] is derived from spin measurements at A and B when detector settings are aligned. If it is correct that when a particle is measured at detector A and is spin up in the y direction , then its entangled twin at B is in superposition...

Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality: ||(x^2)-4|-3|< 1

Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim [x] \geq x - 1 The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an...

Here's a fun problem proof I came across. Show that \left| \frac { z- w }{1 - \overline{z}w} \right| < 1 given |z|<1, |w|<1. I attempted writing z and w in rectangular coordinates (a+bi) but to no avail. Any suggestions, forum?

Prove 2^{\frac{1}{3}}+2^{\frac{2}{3}}<3.

Homework Statement Find δ (which is the input tolerance): lim (9 - 4x2)/(3 + 2x2) = 6 x→-1.5 Homework Equations |f(x) - L| < \epsilon |x - a| < \delta The Attempt at a Solution lim (9 - 4x2)/(3 + 2x2) = 6 x→-1.5 I need to get to: |x-(-1.5)| < \delta = |x + 1.5)| <...

Homework Statement Prove If ## 0 \leq a < b ## and ## 0 \leq c < d ## then ## ac < bd ## The Attempt at a Solution not sure how to even start on this, was thinking if a = 0 or c = 0, then ac = 0, but bd > 0 (which is given) so bd > ac however this seems like I'm cheating because they give...

$a,b,c,d,e,f,g \in N$ $a<b<c<d<e<f<g$ $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}+\dfrac{1}{g}=1$ please find one possible solution of a,b,c,d,e,f,g (you should find it using mathematical analysis,and show your logic,don't use any program)

prove |a+b| \leq |a| + |b| i've proved it considering all the 4 cases for a and b but the book went about it a different way: (|a+b|)^2 = (a+b)^2 = a^2 + 2ab + b^2 \leq a^2 + 2|a||b| + b^2 = |a|^2 + 2|a||b| + |b|^2 = (|a|+|b|)^2 it then goes on the conclude that |a+b| \leq |a| +...

Hello, if you guys would turn to page 117 in Spivak's Calculus, there is the proof for theorem 3. At the last line he stated that this last inequality ##|f(x)-f(a)|<f(a)## implies ##f(x)>0##. How can you check this fact? Can we assume first that ##f(x)-f(a)<0## to eliminate the absolute...

Homework Statement (-3/x) < 3 Homework Equations Dividing/multiplying an inequality causes the inequality sign to change. The Attempt at a Solution I keep getting the wrong solution. I tried two methods. I cannot get the textbook solution (x < -1) Method one: -3 < 3x...

Hi PF Somebody gave me this link could you help me to understand why (9) has to be inserted in (7) when the efficiency \eta < 1?