# Tagged Questions

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### Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$\sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}.$$
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### An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
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### Double sum, find upper bound

I have a double sum $$\sum_{i=1}^{\log(n)} \sum_{j=\log(n) - i}^{\log(n)} \left(\frac{1}{2}\right)^i$$ And I'd like to show it's $\mathcal{O}(1)$ i.e. there is a constant $c$ that is an upper bound of ...
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### Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
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I found a summation identity when I try to figure out the proof of an exercise in Apostol's Mathematical Analysis (Page 27, Exercise 1.26): $\ ~~~~$ If $a_1\geq a_2\geq \dots\geq a_n$ and $b_1\geq ... 1answer 61 views ### Inequality Question about Converging Sum This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (I asked the question before, and there was no answer, so I am asking it again.. I'm not sure ... 2answers 111 views ### Showing that$\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{ ...
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Let $\left(c_{n}\right)_{n},\,\left(d_{n}\right)_{n}$ two successions of complex numbers and let $N$ a large natural number.Is it true that ...
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### Maximum and minimum of weighted sum

For $w_i\ge 0$ and some constants $\alpha_i , i=1,...,n$, what is the maximum and minimum of $\sum_{i=1}^{n}\alpha_i w_i$ subjected to $\sum_{i=1}^{n}w_i=1$? Intuitively, I put all weight on the ...
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### 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 ...
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### Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
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### Show that $\sum_{i=1}^n \frac{1}{i^2} \le 2 - \frac{1}{n}$

So I am able to calculate the given problem and prove $P(K) \implies P(k + 1)$; it's been sometime since I did proofs and I perform my steps I get what Wolfram Alpha shows as an alternate solution. ...
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### Show that if $\frac{x_i}{y_i}<t$ for all $i \leq n$, then $\frac{\sum_{k=0}^{n}x_i}{\sum_{k=0}^{n}y_i}<t$

How to show that if $\frac{x_i}{y_i}<t$ and $x_i,y_i>0$ for all $i \leq n$ , then $$\dfrac{\sum\limits_{k=0}^{n}x_i}{\sum\limits_{k=0}^{n}y_i}<t?$$ I tried the following If ...
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### an inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
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### “point earner” problem: cousin of AM-GM

Let $x(i) > 0$ for $i = 1,2,\cdots,2013$ and $x(1)\cdot x(2)\cdot \cdots \cdot x(2013) =1$ , prove: $$\sum_{i=1}^{2013}x(i)^{2013} \ge \sum_{i=1}^{2013}x(i)^{1/2013}$$
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### 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 ...
### How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$
Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...