All Questions
46
questions
0
votes
1
answer
54
views
Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$
I am trying to understand few of the mathematical steps I have encountered in a paper, there are two of them
(a) $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{...
- 141
0
votes
0
answers
45
views
Inequality with Products and Sums
I need help to find a proof for the following inquality.
Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
0
votes
0
answers
98
views
If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?
Suppose that $$\sum_{i=1}^n x_i \ge a$$
where $a>0$ and $x_i\in (0, b]$ for all $i$. Are there any bounding inequalities we can determine for $$\sum_{i=1}^n \frac{1}{x_i}?$$
I understand that $\...
- 1,615
0
votes
2
answers
93
views
Can someone give me a hint to this question concerning $\sum_{i=1}^n |x-i|$?
Find the smallest positive integer $n$ for which
$|x − 1| + |x − 2| + |x − 3| + · · · + |x − n| \geq 2022$
for all real numbers $x$.
I don't think I can combine any of these terms, right? So I started ...
- 13
3
votes
3
answers
168
views
find a closed form formula for $\sum_{k=1}^n \frac{1}{x_{2k}^2 - x_{2k-1}^2}$
Let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. Order the (real) solutions to $\sqrt{\lfloor x\rfloor \lfloor x^3\rfloor} + \sqrt{\{x\}\{x^3\}} = x^2$ with $x\ge 1$ from smallest to ...
- 2,031
7
votes
1
answer
184
views
Show that $|x_{k+1}-x_k| \leq 1$ (for $0<k<n$) implies $\sum_{k=1}^n |x_k| - \left|\sum_{k=1}^n x_k\right|\leq\lceil(n^2-1)/4\rceil$.
Let $n\ge 1$ be a positive integer and let $x_1,\cdots, x_n$ be real numbers so that $|x_{k+1}-x_k|\leq 1$ for $k=1,2,\cdots, n-1$. Show $$\sum_{k=1}^n |x_k| - \left|\sum_{k=1}^n x_k\right|\leq \left\...
- 1,837
1
vote
1
answer
64
views
If $a_sk^s+a_{s-1}k^{s-1}+...+a_0$ is the basis representation of $n$ with respect to the basis $k$. Then, $0<n\leq k^{s+1}-1$.
If $a_sk^s+a_{s-1}k^{s-1}+...+a_0$ is the basis representation of $n$ with respect to the basis $k$. Then, $$0<n\leq k^{s+1}-1$$.
My attempt:-
By basis represantation, we know that $0\leq a_j<k,...
- 849
1
vote
1
answer
95
views
Prove that $\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$
Let $k,n \in \mathbb{Z}^+$ with $n > 1$. Prove that $$\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$$
I roughly observe that AM-...
- 374
15
votes
2
answers
608
views
The inequality $\,2+\sqrt{\frac p2}\leq\sum\limits_\text{cyc}\sqrt{\frac{a^2+pbc}{b^2+c^2}}\,$ where $0\leq p\leq 2$ is: Probably true! Provably true?
Let $p$ be a positive parameter in the range from $0$ to $2$.
Can one prove that
$$2 +\sqrt{\frac p2} \;\leqslant\;\sqrt{\frac{a^2 + pbc}{b^2+c^2}}
\,+\,\sqrt{\frac{b^2 +pca}{c^2+a^2}}\,+\,\sqrt{\...
- 6,528
6
votes
3
answers
208
views
Algebraic inequality $\sum \frac{x^3}{(x+y)(x+z)(x+t)}\geq \frac{1}{2}$
The inequality is
$$\frac{x^3}{(x+y)(x+z)(x+t)}+\frac{y^3}{(y+x)(y+z)(y+t)}+\frac{z^3}{(z+x)(z+y)(z+t)}+\frac{t^3}{(t+x)(t+y)(t+z)}\geq \frac{1}{2},$$
for $x,y,z,t>0$.
It originates from a 3-D ...
- 1,250
6
votes
5
answers
333
views
Bounds on $S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+\dots+\frac{1}{3001}$
$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$.
Prove that $\dfrac{29}{27}<S<\dfrac{7}{6}$.
My Attempt:
$S<\dfrac{500}{1000} + \dfrac{500}{1500}+ \dfrac{...
user822140
4
votes
1
answer
98
views
Prove the inequality $\sum_{cyc}\frac{a^3}{b\sqrt{a^3+8}}\ge 1$
Let $a,b,c>0$ and such $a+b+c=3$,show that
$$\sum_{cyc}\dfrac{a^3}{b\sqrt{a^3+8}}\ge 1\tag{1}$$
I tried using Holder's inequality to solve it:
$$\sum_{cyc}\dfrac{a^3}{b\sqrt{a^3+8}}\sum b\sum \...
- 93.6k
4
votes
2
answers
124
views
If $a, b, c, d\in\mathbb R^+, $ then prove that $\displaystyle\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b}\ge 0.$
My approach: We have:
$\displaystyle\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b}$
$\displaystyle=\frac{a+c}{b+c}-1+\frac{b+d}{c+d}-1+\frac{c+a}{d+a}-1+\frac{d+b}{a+b}-1$
$\...
5
votes
4
answers
110
views
Prove that for every $n \in \mathbb{N}$ $\sum\limits_{k=2}^{n}{\frac{1}{k^2}}<1$ [duplicate]
$$\sum\limits_{k=2}^{n}{\frac{1}{k^2}}<1$$
First step would be proving that the statement is true for n=2
On the LHS for $n=2$ we would have $\frac{1}{4}$ therefore the statement is true for $n=...
- 191
5
votes
2
answers
364
views
How to find range $a_{75}$ of the term of the series $a_n=a_{n-1}+ {1 \over {a_{n-1}}} $ [duplicate]
If $a_1=1$ and for n>1$$a_n=a_{n-1}+ {1 \over {a_{n-1}}} $$
$a_{75}$ lies between
(a) (12,15)
(b) (11,12)
(c) (15,18)
Now , in this question, I rewrote, $a_n-a_{n-1} = {1 \over {a_{n-1}}}$, to ...
- 1,766
3
votes
3
answers
172
views
Proving $a^2 + b^2 + c^2 \geqslant ab + bc + ca$
Let $a, b, c \in \mathbb{R}$. Show that $a^2 + b^2 + c^2 \geqslant ab + bc + ca$.
My reasoning went as follows and I would like to know if it's correct.
$a^2 + b^2 + c^2 \geqslant ab + bc + ca$
$\...
user713999
5
votes
2
answers
520
views
Prove that $\sum_{i=1}^n\frac{x_i}{\sqrt[nr]{x_i^{nr}+(n^{nr}-1)\prod_{j=1}^nx^r_j}} \ge 1$ for all $x_i>0$ and $r \geq \frac{1}{n}$.
Prove that, for all $x_1,x_2,\ldots,x_n>0$ and $r \geq \frac{1}{n}$, it holds that
$$\sum_{i=1}^n\frac{x_i}{\sqrt[nr]{x_i^{nr}+(n^{nr}-1)\prod \limits_{j=1}^nx^r_j}} \ge 1.$$
This is a slightly ...
- 2,040
2
votes
4
answers
139
views
How do we prove this inequality?
Suppose $a,b,c > 0$. Prove that
$$\frac{a^2}{b^2} +\frac{b^2}{c^2} + \frac{c^2}{a^2} \geq \frac ab + \frac bc + \frac ca.$$
I've tried multiplying everything by the denominator and then I tried to ...
- 1,798
2
votes
2
answers
72
views
Interesting Inequality With Exponents And Base > 1
I had trouble proving the following inequality:
$\beta > 1$
$(\alpha_{1}\beta^{2\alpha_{1}} + \ldots + \alpha_{n}\beta^{2\alpha_{n}})(\beta^{\alpha_{1}} + \ldots + \beta^{\alpha_{n}}) \geq (\...
- 205
0
votes
4
answers
51
views
Prove using squared number property
$$ If \sum_{i=1}^{10} x_i=10 $$ Prove that $$ \sum_{i=1}^{10} x_i^2\ge 10 $$
- 711
1
vote
3
answers
40
views
How do we know that $\sum\limits_{k=1}^n\left(\frac{a}{b}\right)^{k-1}< n $?
I'm trying to convince myself that $$b^n - a^n < (b-a)nb^{n-1}$$ when $0 < a < b$ and $n>1$.
I will be persuaded once I can show that $$\sum\limits_{k=1}^nb^{n-k}a^{k-1}=b^{n-1}\sum\...
- 3,425
2
votes
0
answers
38
views
Upper bound on $\frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}$
Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$...
- 638
0
votes
1
answer
53
views
Need help showing that $0.1\ge \sum_{n=n_r}^{\infty}P(n; \mu=0.85) \implies 0.9 \le \sum_{n=0}^{n_r-1}P(n; \mu=0.85)$
I have the following 'homework style' question which I will quote word for word:
A test is done to check for a predicted atomic spectral line by counting the number of
photons emitted from a ...
- 8,528
2
votes
1
answer
160
views
How to prove that $ \Bigg(\sum^{n}\limits_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}$ for $n\ge1$
I need to prove Prove the inequality
$$ \Bigg(\sum^{n}_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}, $$
where $n$ is a positive integer.
Equivalently
$$\...
- 24.2k
1
vote
1
answer
40
views
simple sum inequality
I have an inequality for sums that I can't proof, although I know it is true.
Let $h_{ij} = h_{ji}$ a real $n\times n$ matrix, and $h_{ijk} = - h_{ikj}$ a real $n\times n\times n$ tensor with $$\sum_{...
- 1,042
0
votes
1
answer
52
views
Smallest Sum of products, under cyclic permutations
Let $a_i$ be real positive numbers, $i=0, \dots, N-1$, with $N$ even.
Define sums $S_k = \sum_{i=0}^{N-1} a_i a_{i+k}$, for $k=0, \dots, N-1$, where the indices are understood to be cyclic, i.e. $...
- 15.8k
0
votes
2
answers
92
views
Find the maximize of $\sum_{cyc}\frac{1}{x^2+y^2+1}$
Let $x>0$, $y>0$ and $z>0$ such that $xy+yz+xz=3$. Find a maximize of $$P=\frac{1}{x^2+y^2+1}+\frac{1}{y^2+z^2+1}+\frac{1}{z^2+x^2+1}$$
We need to prove $P\le 1$ with $x=y=z=1$
We have: $$\...
- 1,898
2
votes
6
answers
1k
views
Showing that $S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000} \gt 1$
If $$S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000}$$ then
$$S\gt 1,$$
but how?
I understood that there are $451$ pair of terms. So clubbed two terms together.
$\frac{1}{100}+\frac{1}{...
- 137
3
votes
2
answers
196
views
Inequality $\frac{1}{a+b}+\frac{1}{a+2b}+...+\frac{1}{a+nb}<\frac{n}{\sqrt{a\left( a+nb \right)}}$
Let $a,b\in \mathbb{R+}$ and $n\in \mathbb{N}$. Prove that:
$$\frac{1}{a+b}+\frac{1}{a+2b}+...+\frac{1}{a+nb}<\frac{n}{\sqrt{a\left( a+nb \right)}}$$
I have a solution using induction, but ...
- 765
3
votes
1
answer
106
views
Inequality $\frac{x_1^2}{x_1^2+x_2x_3}+\frac{x_2^2}{x_2^2+x_3x_4}+\cdots+\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1}+\frac{x_n^2}{x_n^2+x_1x_2}\le n-1$
Show that for all $n\ge 2$
$$\frac{x_1^2}{x_1^2+x_2x_3}+\frac{x_2^2}{x_2^2+x_3x_4}+\cdots+\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1}+\frac{x_n^2}{x_n^2+x_1x_2}\le n-1$$
where $x_i$ are real positive ...
- 765
1
vote
1
answer
116
views
Upper bound of a sum of series
How can I find a tight upper bound for the following expression:
$\sum\limits_{i=1}^{k} a_i \sum\limits_{j = 1}^{i} \frac{1}{b_j} = a_1 \frac{1}{b_1} + a_2 (\frac{1}{b_1} + \frac{1}{b_2}) + \dots + ...
- 41
1
vote
1
answer
180
views
Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\ge\frac{3}{2}$
For $a\geq b\geq c >0$. Prove that $$\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}\ge\dfrac{3}{2}$$
$a=100;b=1;c=1/100$ it's wrong ???
$\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}\ge\dfrac{3}{2}$ ...
- 1,898
2
votes
3
answers
152
views
Inequality related to sum of reciprocals: $\sum_{k=1}^{n} {\frac {1} {k^2}} > \frac {3n}{2n+1}$?
For every integer $n>1$, prove that :
$\sum_{k=1}^{n} {\frac {1} {k^2}} > \frac {3n}{2n+1}$
I don't seem to find any clue on how to relate the left side of the inequality to the right side.
I ...
user399078
6
votes
2
answers
144
views
How to show $\sum\limits_{r=0}^n \frac{1}{r!} \lt\left (1 + \frac{1}{n}\right)^{n+1}$ for all $n \ge 1$?
Using the binomial expansion, it is quite is easy to show that $$\left(1+\frac{1}{n}\right)^n \le \sum_{r=0}^{n} \frac{1}{r!} $$ for all $n\in\mathbb{Z^+}$, with equality holds when $n=1.$ (Can it be ...
3
votes
2
answers
144
views
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
As the title says I need to prove the following by induction:
$$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$
When trying to prove that P(n+1) is true if P(n) is, then I ...
- 706
-4
votes
2
answers
110
views
proving $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ [closed]
how can one prove that:
$\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$
For all natural $n$,
without using induction?
thank you.
- 77
2
votes
2
answers
64
views
Alternate method to prove this series in a better way
Prove that $\frac{1.2 + 2.3 + 3.4 + .....+ n(n + 1)}{n(n + 3)} \ge \frac{n + 1}{4}$ for $n\ge1$
My attempt :
Breaking the series into two different series
$$ S_1 = \sum_{i = 0}^n i^2 = \frac{n(n + ...
- 1,801
7
votes
1
answer
244
views
Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$
If $n > 3$ and $a_1,a_2,\ldots,a_n$ are positive real numbers with $a_1a_2\cdots a_n = 1$, prove that $$\dfrac{1}{1+a_1+a_1a_2}+\dfrac{1}{1+a_2+a_2a_3}+\cdots+\dfrac{1}{1+a_{n-1}+a_{n-1}a_n}+\dfrac{...
- 15.6k
0
votes
1
answer
139
views
Proof that the square root of the mean of the squares is always greater than or equal to the mean of weighted values
I couldn’t think of a better title, but basically you are given some values $x_1$, $\ldots$, $x_n$ and some weights $p_1$, $\ldots$, $p_n$ (with $x_k\in\mathbb{R}$ and $p_k\in[0,1]$, also $p_1+\ldots+...
- 338
2
votes
2
answers
129
views
Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction [duplicate]
Prove by induction that
$1 + \frac {1}{4} + \frac {1}{9} + ... +\frac {1}{n^2} < 2 - \frac{1}{n}$ for all $n>1$
I got up to using the inductive hypothesis to prove that $P(n+1)$ is true but I ...
- 832
1
vote
1
answer
147
views
Upper bound for $\sum_{i=2}^{n} 1/(i\log(i))$
I tried to find the asymtotic upper bounds for these summations
$$\sum_{i=2}^{n} 1/(i\log(i)) \text{ and } \sum_{i=2}^{n} 1/(i\log(i)\log\log(i)) .$$
My guess is that they might be bounded by $O(\log\...
- 783
12
votes
2
answers
937
views
$\sum\limits_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod \limits_{j=1}^nx_j}} \ge 1$, for all $x_i>0.$
Can you prove the following new inequality? I found it experimentally.
Prove that, for all $x_1,x_2,\ldots,x_n>0$, it holds that
$$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod\limits _{j=...
- 2,040
5
votes
3
answers
450
views
Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$
Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$
After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however ...
- 2,916
1
vote
1
answer
109
views
When $(\sum_{i=1}^nk_i < \prod_{i=n}^ni^{k_i}k_i!)$?
Consider $\Omega \subset \mathbb{N}$ a finite subset of $\mathbb{N}$, $\phi: \Omega \rightarrow \mathbb{N}$ an enumeration of $\Omega$ such that $\phi(\omega)=i$ and $|\Omega|=n$,
$$
\sum_{i=1}^n\...
- 4,899
10
votes
3
answers
718
views
Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$
I want ask a question about a sum. The exercise is as follows:
Prove the following inequality for every $n \geq 1$:
$$\sum\limits_{k=1}^n \frac{1}{k^2+3k+1} \leq \frac{13}{20} .$$
- 101
1
vote
3
answers
4k
views
The sum of the reciprocal of the triangular numbers up to $\frac1{t_n}$ is < 2
I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you ...
- 453