# Tagged Questions

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### Estimate on the difference of quotients

The following is supposedly true (found it in a paper), however I fail to see why. Let $L(x)$ be a function that goes to $0$ as $x\rightarrow\infty$, $g(n)$ a sequence which goes to $\infty$ as ...
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Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$... 1answer 60 views ### Using integral estimation to show that  \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4} Show with Integral estimation that$$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}f(k)=\frac {\ln k}{k^2} $$For the integral it is : 1 But the other part is the ... 1answer 46 views ### Two sequences defined by recurrence relations satisfy x_n/y_n<\sqrt{7} for all n Let (x_n)_{n\geq 1} and (y_n)_{n\geq 1} be two sequences such that:$$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$with x_1=2 and y_1=1 Prove that for all n ... 5answers 61 views ### Inequality involving a finite sum this is my first post here so pardon me if I make any mistakes. I am required to prove the following, through mathematical induction or otherwise:$$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + ...
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I read about two inequalities called Chong's inequalities. They state: $$\sum_{k=1}^N\dfrac{a_k}{a_{\pi(k)}}\ge N$$ and $$\displaystyle\prod_{k=1}^Na_k^{a_k}\ge\prod_{k=1}^N a_k^{a_{\pi(k)}}$$ I ...
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### Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
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### Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=−y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
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### Is this sum $S=\sum_{i,j=1}^n\frac{a_ia_j}{i+j}$ greater than or equal to zero?

Given $$(a_1,a_2,...,a_n)\in\mathbb{R}$$ does this inequality hold? $$S=\sum_{i,j=1}^n\frac{a_ia_j}{i+j}\ge0$$ Thanks.
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### How prove concave sequence inequality $\left(\sum_{k=1}^{n}a_{k}\right)^2\ge\frac{3n-c}{4}\sum_{k=1}^{n}a^2_{k}$

let concave sequence $\{a_{n}\}$,such $a)_{n}\ge 0$,and such $$\dfrac{a_{i-1}+a_{i+1}}{2}\le a_{i},i=1,2,\cdots,n-1$$ where $a_{0}=0$. show that $\exists c>0$ such that for every ...
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### Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
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### Inequality $\sum\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum \frac{1}{x + n^2}$

$x\geq0$, then, we have $$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2}$$ The problem is not easy, even $x=1$. Any help will be appreciated
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### Less than or equal summations

Hi,I want to prove above unequal that consist of two summation both of this sides.It a formula in Computer Network to control Congestion.The way to prove it is not important, but because I weak in ...
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### On the numbers divisible by all the Integers not exceeding their $r^{th}$ roots.

Consider the set of all numbers which are divisible by all natural numbers not exceeding their square root, and denote this set by $S_2=\{1,2,3,4,6,8,12,24\}$ (Here the subscript indicates that we're ...
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### Exercise on a series

Prove the following inequality: $$\sum_{k=m+1}^{\infty} \frac{1}{k!}< \frac{1}{m*m!} \forall m\in \mathbb{N^+}$$ My strategy of attack was to set up an inequality like ...
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### sequence $a_n = \lceil \sqrt{2}n \rceil$

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$ Direct proof I tried but could not figure out. I tried fixing m ...
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### Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
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### Proving an inequality for a sequence by induction

I'm having some trouble with the following problem: Let $a_n$ be a sequence defined iteratively for $n \geq 0$ as follows: $a_n = a_{m+1} + 2a_m + a_{n-m-1} + 2$ where $m$ is defined as ...
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 ...
for $x > -1$, Prove the following inequality: $$\left( {\ln (1 + x) + \sum\limits_{k = 1}^n {\frac{{{{( - 1)}^k}{x^k}}}{k}} } \right){( - x)^{n + 1}} \le 0$$ Following the advice to use ...