4
votes
2answers
71 views

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

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
1
vote
0answers
19 views

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 ...
0
votes
2answers
40 views

How to derive these inequalities?

I can derive the inequalities $$ n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p $$ for any positive integers $p$ and $n$. These actually follow from the identity $$b^p - a^p = (b-a)(b^{p-1} + ...
2
votes
2answers
19 views

How to obtain this upper bound on the summation from this inequality?

I can show that $$ \frac{1}{\sqrt{n}} < 2 (\sqrt{n} - \sqrt{n-1} ) $$ for $n \geq 1$. Now from this how to derive the following inequality? $$ \sum_{n=1}^m \frac{1}{\sqrt{n}} < 2\sqrt{m} - 1 $$ ...
0
votes
1answer
39 views

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 ...
0
votes
2answers
38 views

Inequation of an sum smaller than 1

I'm trying to figure out the following $$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$ How I can show it if $q \geq 2$? Maybe with telescoping sums? Thanks, Landau
2
votes
4answers
66 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
42 views

Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? \begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
1
vote
1answer
36 views

How to prove lower and upper bound for exponential sum?

A paper I'm reading implicitly uses the fact $$\sum\limits_{t=1}^n e^{-ta^2} \in \theta(\frac{1}{a^2})$$ (It uses the both $\leq$ and $\geq$ sides in the proofs). I'm able to prove that ...
1
vote
1answer
49 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
1
vote
2answers
48 views

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 ...
0
votes
0answers
40 views

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 ...
3
votes
2answers
76 views

A summation identity which is for me hard to verify

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 ...
0
votes
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 ...
3
votes
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{ ...
0
votes
1answer
30 views

About complex sum

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 ...
2
votes
1answer
47 views

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 ...
0
votes
1answer
44 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 ...
3
votes
1answer
83 views

Cauchy-Schwarz inequality on double-summation term

I have the following, where $v$ is a vector $$ v\cdot (v\cdot \nabla)v $$ which in index notation becomes $v_jv_id_iv_j$. I want to apply the Cauchy-Schwarz inequality on this, which is given by $$ ...
7
votes
4answers
323 views

Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\ldots+\frac{1}{3001}<\frac43$

Prove that $$1<\dfrac{1}{1001}+\dfrac{1}{1002}+\dfrac{1}{1003}+\ldots+\dfrac{1}{3001}<\dfrac43 \, .$$ My work: $$\begin{eqnarray*} ...
0
votes
3answers
74 views

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 ...
18
votes
4answers
302 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
2
votes
1answer
38 views

Is $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ for all integers $j$ and $k$, where $0\leq j\leq k$?

For all integers $j$ and $k$, where $0\leq j\leq k$, is the inequality $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ true? I have a feeling that it is and it would be helpful to me if it is, ...
0
votes
1answer
80 views

Adding/multiplying summations of different indices to prove Cauchy-Schwarz

$\sum_{i=1}^{n}\sum_{j=1}^{n}(a_ib_j-a_jb_i)^2 = \sum_{i=1}^{n}a_i^2\sum_{j=1}^{n}b_j^2 + \sum_{i=1}^{n}b_i^2 \sum_{j=1}^{n}a_j^2 - 2\sum_{i=1}^{n}a_ib_i \sum_{j=1}^{n}b_ja_j$ Can someone walk me ...
2
votes
1answer
29 views

Upper bound on a sum similar to a telescoping sum: $|p_{n'}-p_0| \leq \sum_{i=1}^{n'}{|p_{i}-p_{i-1}|}$

Does someone know why the following is true: $$|p_{n'}-p_0| \leq \sum_{i=1}^{n'}{|p_{i}-p_{i-1}|}$$ If we did not have the "size of" operator, there would be an equality, but in this case its a ...
1
vote
0answers
146 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
3
votes
2answers
284 views

Induction: show that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$

The question: show by using induction that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$ My attempt at a solution: The base case $n = 1$ is true. First we use the ...
1
vote
0answers
54 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
0
votes
1answer
57 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
1
vote
3answers
114 views

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. ...
1
vote
2answers
38 views

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 ...
11
votes
3answers
548 views

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 ...
0
votes
2answers
118 views

“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: $$ \sum_{i=1}^{2013}x(i)^{2013} \ge \sum_{i=1}^{2013}x(i)^{1/2013} $$
5
votes
2answers
74 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 ...
5
votes
0answers
109 views

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: ...
1
vote
1answer
59 views

What is the smallest possible weighted average of 1…n?

Let $a_1, ..., a_n$ be numbers in the range $[{1 \over n},1]$. Define: $$ U = \sum_{i=1}^{n}{a_i} $$ $$ W = \sum_{i=1}^{n}{i\cdot a_i} $$ I am looking for the largest possible value of the ratio $U ...
8
votes
2answers
146 views

How find this value $\inf_{n\ge 1}\left(\min_{x\in[0,\frac{\pi}{2}]}\left(\sum_{k=1}^{n}\frac{\cos{(kx)}}{k}\right)\right)$

Find this $$\inf_{n\ge 1}\left(\min_{x\in[0,\dfrac{\pi}{2}]}\left(\sum_{k=1}^{n}\dfrac{\cos{(kx)}}{k}\right)\right)$$ I know this Young inequality ...
3
votes
2answers
83 views

Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$ [duplicate]

It basically says it all in the title. I tried solving the inequality using the bernoulli inequality somehow $$\dfrac{\displaystyle\sum_{k=0}^n\frac{1}{k!}}{(1+\frac{1}{n})^n}\geq 1,$$ but the ...
0
votes
0answers
35 views

Poisson probability inequality

How do I find $S$ such that the inequality $CL \le e^{-n\lambda R}\sum_{k=0}^S \frac{(n \lambda R)^k}{k!}$ holds, where $n$ is a positive integer representing number of units in service, $\lambda$ is ...
0
votes
0answers
45 views

Proving a Certain Inequality that Involves the Sinc Function

Could someone kindly show me how to rigorously prove that there exists a constant $ C > 0 $ such that $$ \forall N \in \mathbb{N}: \quad \sup_{x \in \mathbb{R}} \sum_{\substack{k \in \mathbb{Z} \\ ...
6
votes
2answers
133 views

Simple Divisor Summation Inequality (with Moebius function)

Show that $$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$ where $\mu$ is Moebius function and n is a positive integer. The hard thing here is that the sum is not directly ...
0
votes
2answers
64 views

Find an index n such that inequality is true

I need to find an index $n$ such that: $|e^{x} - S_{n}(x)| \leq \frac{|e^{x}|}{10^{4}}$ (1), where $S_{n} = \sum\limits_{i=0}^{n} \frac{x^{k}}{k!}$ is the n-th Partial Sum of $e^x$. Let $x$ be a ...
5
votes
1answer
84 views

$\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions. Question 1 : Is the following true? $$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$ Question 2 : Is the ...
0
votes
1answer
96 views

How prove this$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^my^j=1$

let $m,n$ be positive numbers,and $x,y>0$ such $x+y=1$,show that $$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^jy^m=1$$ My try: ...
2
votes
2answers
197 views

$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$

$\displaystyle\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$ Note that $\forall x\ge 0, \sqrt{x}-1\le\sqrt{1+x}-1\le x$ Then ...
0
votes
1answer
63 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
2
votes
2answers
230 views

Inequality $\sum^{k}_{i=1}\sum^{n}_{j=k}{{1}\over{j - i + 1}} \le n$

I was proving a statement and have finished all the other part, the remaining boiling down to proving the inequality below (if I proved the other part correctly): ...
0
votes
0answers
61 views

Gronwall inequality discrete version difference form

I have a question from my homework regarding Gronwall inequality: Given $\Delta t,\gamma,$ and $a_n, b_n, c_n$ sequences of nonnegative numbers for $n\geq0$ Define $f_0:=a_0$. Suppose the inequality ...
2
votes
3answers
51 views

inequality question on real numbers #2

Expirimentally it seems that $$\sum_{1\leq j\leq \lfloor n/8 \rfloor}\left(\frac{\pi en}{4j}\right )^{j/2}<2^{c_0n}$$ where $c_0=0.6$ and large $n$. Is there any proof? Thank you.
0
votes
1answer
64 views

Inequality with a sum and factorial

For a homework assignment we have the following question that I'm stuck on. Let $ 0 \leq y \leq 1 $ be given. $\forall m \in \mathbb{N}$, define $ \displaystyle S_m(y)=\sum_{k=0}^m \binom{m}{k}y^k$. ...