All Questions
56
questions
-3
votes
2
answers
155
views
Prove : $a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$ [closed]
Prove that by Cauchy inequality:
$a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$
$a,b,c$ are a positive numbers
$a*b*c=1$
- 33
2
votes
1
answer
100
views
How to prove that for fixed number $m$ positive numbers the sequence $\frac{\sum\limits_{k=1}^ma_k^{n+1}}{\sum\limits_{k=1}^ m a_k^{n}}$ is monotone?
I saw this question in my book
Let $a_1 , a_2 , \dots,a_m$ be fixed positive numbers and $S_n =\frac{\sum \limits_{k=1}^ m a_k^{n}}{m}$ Prove that $\sqrt[n] {S_n} $ is monotone increasing sequence
...
- 6,620
6
votes
2
answers
101
views
Prove that $\sum_{i=1}^{n} (x_i-\overline{x}_n)^4 \leq \sum_{i=1}^{n} x_i^4$
I'm trying to prove the following inequality:
$\sum_{i=1}^{n} (x_i-\overline{x}_n)^4 \leq \sum_{i=1}^{n} x_i^4$
where $\overline{x}_n=\frac{1}{n}\sum_{i=1}^{n}x_i$.
This is my attempt: we can easily ...
- 61
5
votes
3
answers
439
views
Proving the inequality with finite sums
I would like to show the following inequality involving finite sums ($x>0$):
$$2 \left(\sum_{n=1}^k nx^n\right)^2 + \sum_{n=0}^k x^n \sum_{n=1}^k nx^n - \sum_{n=0}^k x^n \sum_{n=1}^k n^2x^n \geq 0 ...
user823777
2
votes
1
answer
57
views
prove that the maximum value of a function is attained when all $x_i$'s are 0 or 1
Let $0\leq x_i\leq 1$ for $1\leq i\leq n.$ Prove that the maximum value of the sum $S(x_1,\cdots, x_n) = \sum_{i=1}^n x_i - \sum_{i=1}^n x_i x_{i+1},$ where $x_{n+1} := x_1$ is attained when all $x_i$'...
- 1,137
2
votes
1
answer
47
views
prove the maximum of $\sum_{1\leq r<s\leq 2n}(s-r-n)x_rx_s$ is attained when all $x_i$'s are $-1$ or $1$
Let n be a positive integer. Prove that the maximum possible value of $Z =\sum_{1\leq r<s\leq 2n}(s-r-n)x_rx_s$, where $-1\leq x_i\leq 1$ for all i, is attained when $x_i\in \{-1,1\}\,\forall i$.
...
- 1,837
0
votes
1
answer
84
views
Inequality involving finite sum and integral
I'm reading a proof where they use the following inequality:
$$\sum_{k=4}^n \frac{k^2}{n}(1-a)^{k+1}\le\int_0^\infty \frac{x^2}{n}\exp{(-ax)}$$
For $a>0$. I'm trying to show it.
So far I got
$$\...
- 376
1
vote
2
answers
93
views
Prove that $|\sum\limits_{k=1}^{n} a_{k}| \ge |a_{1}| - \sum\limits_{k=2}^{n} |a_{k}|$.
I am abjectly disappointed that I could not prove this statement on my own. I have tried it directly and by contradiction but hit a wall. Here is the statement (again) and my proof (thus far):
Prove ...
- 43
5
votes
2
answers
240
views
Proving $\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+1}{n+1})}k>\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+3}{n+1})}k$
A few days ago I asked a question about an interesting property of the partial sums of the series $\sum\sin(nx)/n$.
Here's the link:
Bound the absolute value of the partial sums of $\sum \frac{\sin(nx)...
- 1,684
12
votes
4
answers
562
views
Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$
Problem: Let $x_1, x_2, \cdots, x_n$ ($n\ge 2$) be reals. Find upper bounds for
$$\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}. $$
...
- 40.3k
1
vote
0
answers
129
views
Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$
Problem: Find Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$.
This is the follow up of Prove that $|\sin 1| + |\sin 2| + |\sin 3| +\cdots+ |\sin 3n| > 8n/5$.
My attempt
By the ...
- 40.3k
-2
votes
1
answer
126
views
How can I prove $\sum_{i=1}^n \sum_{j=1}^n \frac{a_ia_j}{i+j-1}$ is never negative for any set of n real numbers $a_i$ [closed]
I can't figure this out, can someone help me prove it? I know you guys will come up with an incredibly elegant solution
- 1
0
votes
2
answers
51
views
How get from $\sum_{i=0}^{n-1}\left(1+\frac{1}{n}+\frac{1}{4n^2}\right)^i\left(1+\frac{1}{n}\right)^{n-1-i}$ this $n\left(1+\frac{1}{n}\right)^{n-1}$
in that topic Chen Jiang(second answer) show how to prove $\frac{1}{4n}< e -\left(1 + \frac{1}{n} \right)^n$
I dont understand how, he do that
$\sum_{i=0}^{n-1}\left(1+\frac{1}{n}+\frac{1}{4n^2}\...
- 45
2
votes
1
answer
483
views
An inequality for series with fractional exponent
Let $\{a_k\}_{k\in\mathbb{Z}}$ be a bounded and non negative sequence in $\mathbb{R}$, suppose that there exists $N\in\mathbb{Z}$ such that: $a_k=0$, $\forall k\geq N$. Let $p\in(0,1)$. Is true that:
$...
- 365
1
vote
3
answers
71
views
Bounding sum by (improper) integral
I am trying to verify the following inequality that I came across while reviewing some analysis exercises online:
$$
\sum_{n=1}^{k} \left(1-\frac{n}{k}\right)n^{-1/7}\leq \int_{0}^{k}\left(1-\frac{x}{...
- 35