2
votes
4answers
57 views

Proving for all n that $\sum_{i=0}^n \frac1{2^{i}} < 2$

Proving for all n $\in \mathbb N$, $$\sum_{i=0}^n \frac1{2^{i}} < 2$$ Hint. First prove that the left hand side can be expressed in closed form, i.e. without using the summation operator. This is ...
0
votes
2answers
67 views

How to solve this inequality?

I have the following problem in an assignment and have been struggling to do it. $2 + 2x - x^2 \geq 2 \sqrt{1+2x}$ I have tried solving for $x$ but have not been able to do so. Any hints to solve ...
6
votes
4answers
80 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
1
vote
1answer
16 views

2 Linear equation problems [closed]

Write objective, constraints and graph for the following two problems: 1.A test offers 2 types of problems. Type A takes 3 Min to solve and B takes 2. You have 20 min to take the test and can only ...
0
votes
1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
1
vote
1answer
24 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
0
votes
4answers
63 views

Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
0
votes
1answer
45 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
2
votes
1answer
71 views

To prove $2^{n(n+1)} >(n+1)^{(n+1)} \prod\limits_{j=1}^n \left(\dfrac {j}{n+1-j}\right)^j , \forall n\in \mathbb N $ \ { $1$ }

How do we prove that $2^{n(n+1)} >(n+1)^{(n+1)} \prod_{j=1}^n \Bigg(\dfrac {j}{n+1-j}\Bigg)^j , \forall n\in \mathbb N$ \ {$1$} ?
2
votes
3answers
123 views

Finding the values of $A,B,C,D,E,F,G,H,J$

Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and $$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$ $$C = B + 1$$ $$H = G + 3$$ find (edit: ...
1
vote
4answers
178 views

Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$

The question statement from my homework booklet goes: Prove by mathematical induction that $n^{n+1} > n(n+1)^{n-1}$ is true for all integers $n \geq 2$. I've managed to come up with this ...
1
vote
2answers
74 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
2
votes
2answers
119 views

Solve inequality: $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$

Rational method to solve $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$ inequality? I tried to lead fractions to a common denominator, but I think that this way is wrong, because I had ...
3
votes
12answers
323 views

How to prove this inequality $ x + \frac{1}{x} \geq 2 $

I was asked to prove that: $$x + \frac{1}{x}\geqslant 2$$ for all values of $ x > 0 $ I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a ...
0
votes
3answers
80 views

Solving $x^2 - 16 > 0$ for $x$

This may be a very simple question, but I can't figure out the correct reason behind it. If $x^2 - 16 >0$, which of the following must be true? a. $4 < x$ b. $-4 > x > 4$ ...
3
votes
2answers
104 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ ...
3
votes
1answer
92 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
0
votes
1answer
37 views

Inequality in non-decreasing sequence

Let $a, b$ be two sequences of real numbers such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Prove (or disprove) that ...
2
votes
1answer
36 views

Prove that $P[X>\epsilon] \leq M(t)/e^{\epsilon t}$

Prove that $P[X>\epsilon] \leq \dfrac{M(t)}{e^{\epsilon t}}$ Looks like Markov's inequality, it's very easy to derive for $t>0$ $P[X>\epsilon] =P[Xt>\epsilon t]=[e^{Xt}>e^{\epsilon ...
5
votes
1answer
108 views

Inequality related with concave property

Assume that $f>0,f'<0$ and $f$ is logconcave(the log of $f$ is concave) and twice differentiable. Can we prove, or give a counter example to the following claim: there exists $\bar x>0$ such ...
5
votes
6answers
103 views

How do I prove $\sqrt{x^2 + y^2} \le |x| + |y|$?

Only a hint on how to prove this, if not a complete proof, would also be appreciated.
2
votes
1answer
165 views

Logarithms melting my brain

So I've got an inequality: $\ln(2x-5) > \ln(7-2x)$ and I attempt to solve by doing the following: $$\frac{\ln(2x)}{\ln(5)} > \frac{\ln(7)}{\ln(2x)}$$ $$\Rightarrow \ln(2x) \cdot \ln(2x) > ...
0
votes
1answer
51 views

Writing an inequality and graphing it on a number line

So lets say, the student council set a goal of raising \$200 in cookie sales and it only raised \$100 so far. Write an inequality to show how many more dollars, d, the student council needs to reach ...
1
vote
0answers
34 views

Proving an inequality involving integral

Let $g: [a,b]\mapsto [0,1]$, with $\int_a^b|g'(t)|^2\,\mathrm{d}t\leq 1$. Suppose $b-a<\delta$, and define $$ \bar{g}=\frac{\int_{a}^{b}g\left(t\right)\,\mathrm{d}t}{b-a} $$ Show for ...
1
vote
3answers
69 views

Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$

$a;b;c\in \mathbb{R}^+$ such that $a+b+c=6$. Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$ Thanks :) I have no ideas about this problem ! :(
4
votes
1answer
82 views

Prove: $\frac{2x^2+xy}{(y+\sqrt{zx}+z)^2}+\frac{2y^2+yz}{(z+\sqrt{xy}+x)^2}+\frac{2z^2+zx}{(x+\sqrt{yz}+y)^2}\ge1$

Prove: $$\frac{2x^2+xy}{(y+\sqrt{zx}+z)^2}+\frac{2y^2+yz}{(z+\sqrt{xy}+x)^2}+\frac{2z^2+zx}{(x+\sqrt{yz}+y)^2}\ge1$$ ($x,y,z>0$)
1
vote
3answers
43 views

A question on inequalities

What is the solution set of the inequality $$ \frac{2x - 1 }{x+1}\lt0$$ One answer that is quite simple to get is $$x\lt1/2 $$ What can be the other value for the solution set...??
1
vote
4answers
80 views

Showing $a \le b$ if $a \le b+\varepsilon$, for all $\varepsilon \gt 0$

So I think this is the last problem I have and I'm not thinking I'm doing it properly. Let $a,b$ be real numbers and suppose for all $\varepsilon \gt 0, a \le b+\varepsilon$. Show that $a \le b$. ...
2
votes
1answer
38 views

proving if $0 \le a \le \varepsilon$ for all $\varepsilon \gt 0$ then $a=0$

Suppose $a$ is a real number and we know that $$0 \le a \le \varepsilon$$ for every $\varepsilon \gt 0$. I need to show that $a=0$. The book I am working out of already has shown by contradiction ...
4
votes
2answers
83 views

If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$

If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$. This seems like a basic proof and I think this is how it goes: $$c \le d, \text{ Given }$$ $$a+c \le a+d$$ $$a+c \lt b+d, \text{ since } a \lt ...
1
vote
0answers
73 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
0
votes
1answer
30 views

How to get this inequality using induction (analysis)

Consider the following functions $\theta:\Bbb R\to\Bbb R$ and $\Theta:\Bbb R^n\to\Bbb R$, sucha that: $$ \theta(x) := \begin{cases} 1-|x| & \text{if $|x|\le1$} \\ 0 & \text{if $1\le|x|$} \\ ...
0
votes
2answers
45 views

Find : $\min P=2x^2-xy-y^2$?

$x;y\in \mathbb{R}$ such that : $x^2+2xy+3y^2=4$. Find : $\min P=2x^2-xy-y^2$ ? Thanks :) P/s : I have no ideas about this problem ! :(
0
votes
6answers
107 views

Show if $a^2+b^2 \le 2$ then $a+b \le 2$

If $a^2+b^2 \le 2$ then show that $a+b \le2$ I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without ...
2
votes
1answer
68 views

Cauchy-Schwarz-like inequality of integrals

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$ I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$ so if we ...
0
votes
0answers
38 views

Simple absolute value inequality proof

Prove that if $|x-x_0| < $ $\frac{\epsilon}{2}$ and $|y-y_0| < $ $\frac{\epsilon}{2}$ , then $|(x+y)-(x_0+y_0)|$ $< \epsilon$ and $|(x-y)-(x_0-y_0)|$ $< \epsilon$.
3
votes
0answers
63 views

Find the minimum value of: $P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$

Let $a,b,c\ge0$ such that: $(a+b)(b+c)(c+a)=1$. Find the minimum value of: $$P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$$. I've tried many things but all failed. Please ...
1
vote
1answer
57 views

How prove this inequality: $\frac{x^3+1}{\sqrt{x^4+y+z}}+\frac{y^3+1}{\sqrt{y^4+z+x}}+\frac{z^3+1}{\sqrt{z^4+x+y}}\geq2\sqrt{xy+yz+zx}$

Let $x,y,z>0$ such that $xyz=1$. Show that: $$\dfrac{x^3+1}{\sqrt{x^4+y+z}}+\dfrac{y^3+1}{\sqrt{y^4+z+x}}+\dfrac{z^3+1}{\sqrt{z^4+x+y}}\geq2\sqrt{xy+yz+zx}$$ I've tried many things but all failed. ...
0
votes
2answers
39 views

Probability inequality proof

I'm stuck on a homework question and don't even know where to start. Here it goes: If A and B are two events which are not impossible, prove that $$P(A\land B)\times P(A\lor B)\le P(A)\times P(B)$$
2
votes
1answer
55 views

Triangle Inequality Question

If $-2 \leq x \leq \pi/2$, show that $|2x^3 - 4x^2 + 3x - \sin x|\leq 39$ can someone help me with this question? i'm having difficulty trying to incorporate triangle inequality with it.
1
vote
1answer
121 views

A hard inequality with $a+b+c=3$

Let $$\displaystyle\left\{ \begin{array}{l} a,b,c>0\\ a+b+c=3 \end{array} \right.$$ Prove that: ...
2
votes
2answers
92 views

Show that $d^\ast$ is a metric

For $x$ and $y$ in $R$, let $d(x,y)$ be a metric. Show that $$d^\ast(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is also a metric. It is fairly straightforward to show that $d^\ast(x,y)=0$ if $x=y$ ...
0
votes
2answers
44 views

Proving that $f(p)= \sum_{j \neq i}p_ip_j$ is concave

Let $p_i \in [0,1], 1 \leq i \leq n$ so that $p_1+p_2 +...+p_n=1$. Define $f(p)= \sum_{j \neq i}p_ip_j=1-\sum_{i}p_i^2$. Prove that $f(p)$ is concave. My effort: Let $a,b \geq 0, a+b=1$. I try to ...
-2
votes
1answer
42 views

Min $A=14(a^{2}+b^{2}+c^{2})+\frac{ab+bc+ac}{a^{2}b+b^{2}c+c^{2}a}$?

$a;b;c\in \mathbb{R}^+$ such that $a+b+c=1$. Find the minimum of $A=14(a^{2}+b^{2}+c^{2})+\frac{ab+bc+ac}{a^{2}b+b^{2}c+c^{2}a}$
-1
votes
1answer
54 views

x^2+y^2+z^2=a then what is the of range of xy+yz+zx [closed]

$$x^2+y^2+z^2=a$$ then what is the of range of $$xy+yz+zx$$ options A) $[-a, a]$ B) $[-a/2, a/2]$ C) $[-a/2, a]$
0
votes
2answers
130 views

Find max: $\frac{a}{b+2a}+\frac{b}{c+2b}+\frac{c}{a+2c}$

For $a,b,c>0$ and $abc=1$. Find max: $\frac{a}{b+2a}+\frac{b}{c+2b}+\frac{c}{a+2c}$
2
votes
2answers
118 views

Find max: $M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}$

For $a,b,c>0$ and $abc=1$. Find max: $M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}$
2
votes
1answer
67 views

Find max: $\frac{1}{a^3+2b^3+6}+\frac{1}{b^3+2c^3+6}+\frac{1}{c^3+2a^3+6}$

For $a,b,c>0$ and $abc=1$. Find max: $\frac{1}{a^3+2b^3+6}+\frac{1}{b^3+2c^3+6}+\frac{1}{c^3+2a^3+6}$ I used AM-GM for $a^3+2b^3$ but I don't know how to continue ...
0
votes
0answers
65 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
5
votes
2answers
68 views

Proving a lower bound on the limit superior of a sequence.

Prove that for every positive sequence {$a_{n}$}, $$\varlimsup_{n \to \infty}\frac{\sum_{i=1}^{n+1}a_{i}}{a_{n}}\geq 4$$ Also find the sequences {$a_{n}$} for which 4 is attained. Attempted ...