-2
votes
1answer
28 views
-3
votes
0answers
55 views

Which of the following is correct?

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
4
votes
0answers
31 views

Finite Series Inequality [duplicate]

For each $n=1,2,{\dots}$ and $x\in(0,{\pi})$, prove that the series $$S_n(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k}>0$$
0
votes
1answer
41 views

Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
2
votes
1answer
186 views

if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$

Question: Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall ...
0
votes
1answer
18 views

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 ...
0
votes
2answers
30 views

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 ...
8
votes
3answers
119 views

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 ...
0
votes
1answer
59 views

Very complicated limit and trying to find convergence

I have no idea how to prove this: $\lim_{n \to \infty} \frac{2^2 \times4^2\times6^2\times\dots\times(2n)^2}{(1\times3)(3\times5)\dots((2n-1)(2n+1))} = \lim_{n \to \infty}\frac{2^2 ...
1
vote
2answers
72 views

$e^x>1+(1+x)\log(1+x)$,for $x>0$ using infinite series.

$e^x>1+(1+x)\log(1+x)$,for $x>0$ using infinite series. Attempt $\sum x^k(\frac{1}{k!}+\frac{(-1)^{k+1}}{k!}+\frac{(-1)^{k+1}x}{k!})>1$. any hints?
0
votes
1answer
59 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 ...
0
votes
2answers
214 views

Waring's Inequality Solution

$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$ We craft an inequality, with $m,n ...
0
votes
1answer
67 views

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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
53 views

Prove an inequality

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 ...
0
votes
1answer
63 views

Find an N satisfying this inequality

I am trying to catch up again. $$ \frac{1}{2^{n}} + \frac{1}{3^{n}} + \frac{1}{4^{n}} < \frac{1}{365} $$ Find an $N$ whereby all $n \geq N$ give correct outcomes I thought that as long as ...
2
votes
1answer
55 views

Show that $\lim \inf a_n\le\lim\inf s_n.$

Let $\{a_n\}$ be a bounded sequence of real numbers. Let $s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N.$ Show that $\lim \inf a_n\le\lim\inf s_n.$ The only definition I know of limit ...
0
votes
0answers
18 views

Convergence of the sequence $v_n$ governed by $$v_{n+1}^2\le (1+\alpha_n)v_n^2+\gamma_nv_n-u_n+\beta_n,\quad n\ge 1$$

My question is that if there is a sequence of positive reals $v_n$ which is governed by the following inequalities $$v_{n+1}^2\le (1+\alpha_n)v_n^2+\gamma_nv_n-u_n+\beta_n$$ where ...
0
votes
0answers
30 views

Inequalities and sequences

I ran across this problem in Gelfands Algebra today. 1/2 < 1/101 + 1/102 + ... + 1/200 < 1 He shows proof that this is true 1/2 < 1 - 1/2 + 1/3 - 1/4 + ... + 1/199 - 1/200 < 1 Again he ...
1
vote
1answer
74 views

the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
1
vote
1answer
51 views

How prove this $\frac{1}{b_{k+1}b_{k+2}}+\frac{1}{b_{k+2}b_{k+3}}+\cdots+\frac{1}{b_{2k}b_{2k+1}}>\frac{1}{12345},$

let sequence $\{b_{n}\}$,and $b_{n}>0$,let $$S_{n}=b_{1}+b_{2}+\cdots+b_{n}\le n^{\frac{3}{2}},\forall n\ge 1$$ show that ...
10
votes
3answers
287 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any poistive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
1
vote
0answers
45 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 ...
1
vote
0answers
62 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
1
vote
0answers
22 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
0
votes
2answers
53 views

solving or showing an inequation [closed]

could somebody show me this relationship and also explain why I have to take the required steps: $\vert x+y \vert$ $\leq$ $\vert x\vert +\vert y\vert$ for all real numbers x,y $\in \mathbb{R}$ Thank ...
1
vote
0answers
67 views

Concerning the sequence $ x_1=1 ,\space x_{n+1}=x_n+\dfrac{1}{x_n} $ [duplicate]

Let $(x_n)$ be a real sequence satisfying $ x_1=1 ,\space x_{n+1}=x_n+\dfrac{1}{x_n} ,\forall n \in \mathbb N$ , then can we find an expression for $[ x_n$] ( box-function) in terms of $n$ $,\forall ...
2
votes
2answers
35 views

Inequality from a Sequence

I am working on a problem and I am lead to prove the following inequality which is true based on writing out the sequence and on the fact that it should be true based on what I am trying to prove. ...
5
votes
2answers
68 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 ...
2
votes
1answer
65 views

Range of inner product of a sequence and its permutation

$a^n :=(a_i)_1^n$ is a finite sequence of real numbers of length $n$, where $\sum\limits_{i=1}^n a_i=0$ and $\sum\limits_{i=1}^n a_i^2=1$. Consider $s_n(a^n,\sigma):=\sum\limits_{i=1}^n ...
1
vote
0answers
69 views

necessary and sufficient conditions for the following inequality to be hold

Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ and let $t\in(0,1)$. find necessary and sufficient ...
5
votes
2answers
99 views

Show that the sum of the series

Show that the sum of the series is greater than 24 $$\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{9}+\sqrt{11}} +\cdots+\frac{1}{\sqrt{9997}+\sqrt{9999}} > 24$$ I see ...
5
votes
0answers
104 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: ...
6
votes
1answer
299 views

The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent

If a series $\sum\limits_{n=1}^\infty a_n$ is convergent, and $a_n\gt0$... Do not refer to Carleman's inequality or Hardy's inequality, show that the series $$\sum_{n=1}^\infty \frac ...
8
votes
3answers
103 views

Sum $S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$

Consider the sum $$S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$$ where $0\le c\le 1$. When $c=0$, $S(n,c)$ grows asymptotically as $n\log n$. When $c=1$, $S(n,c)$ grows asymptotically as $n$. ...
2
votes
1answer
80 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
1
vote
1answer
70 views

How prove this sequence inequality $a_{n}+a_{n+1}<\frac{5}{2}\sqrt{n}$

Question: let sequence $\{a_{n}\}$ such $a_{1}=1$,and $$a_{n+1}a_{n}=n$$ show that $$a_{n+1}+a_{n}<\dfrac{5}{2}\sqrt{n}$$ My try: since $$a_{1}a_{2}=1\Longrightarrow a_{2}=1$$ so ...
5
votes
4answers
101 views

Show that $|\sin\frac{1}{n^2}|<\frac{1}{n^2}$, $n=0, 1, 2, \dots$

As part of showing that $$ \sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right| $$ converges, I ended up with trying to show that $$ ...
1
vote
2answers
33 views

Reasoning why the implication $t - \epsilon \le x \le t + \epsilon$ for $\epsilon \ge 0 \Rightarrow x = t$ holds using sequences.

In texts I've seen the following reasoning used several times: Suppose $t - \epsilon \le x \le t + \epsilon$ holds for $\epsilon \ge 0$. Then it in particular holds for $t - \frac 1 n \le x \le t + ...
4
votes
2answers
89 views

How prove this $a_{n}>1$

let $0<t<1$, and $a_{1}=1+t$, and such $$a_{n}=t+\dfrac{1}{a_{n-1}}$$ show that $a_{n}>1$ My try: since $$a_{1}=1+t>1$$ ...
2
votes
2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
0answers
69 views

What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
0
votes
2answers
72 views

How to prove this inequality sequence?

${b_{n}},b_{1}=1,b_{n}b_{n+1}=\dfrac{1}{(2n+1)}$,where $n \geq 3, n \in N_{+}$ Show that : $ 1. b_{2n} < b_{2n+1} < b_{2n-1} $ ; $ 2. b_{1} + b_{2} + b_{3} + \cdots + b_{n} > \sqrt{2n + ...
0
votes
1answer
37 views

about function series

Good evening, everyone, Could anyone please tell me how to check if the series $\sum_{k\geq 2}\dfrac{1}{k^4+x}$ is greater than $C\sum_{k\geq 2}\dfrac{1}{k^2+x}$ where $C$ is independent of the ...
3
votes
3answers
73 views

How to prove the inequality $\sum_{i=1}^n \frac{\sqrt{i+1}}{2i} > \frac{\sqrt{n}}{2} $ for $n\in\mathbb{Z}^+$?

I have to prove this inequality: $$ \forall n \in Z^+, \sum_{i=1}^n \frac{\sqrt{i+1}}{2i} > \frac{\sqrt{n}}{2} $$ So far, I have done the base cases and assumed the inequality is true for some ...
6
votes
1answer
150 views

About $e^{\pi}\gt {\pi}^e, \ e^{e^{\pi}}\lt {\pi}^{{\pi}^{e}},e^{{\pi}^{{\pi}^e}}\gt {\pi}^{e^{e^{\pi}}}$ and their generalization

Let us define a sequence $\{(a_n,b_n)\}$ as $$(a_1,b_1)=(e,\pi),\ \ (a_{2n},b_{2n})=(e^{b_{2n-1}},{\pi}^{a_{2n-1}}),\ \ (a_{2n+1},b_{2n+1})=(e^{a_{2n}},{\pi}^{b_{2n}})$$ for $n=1,2,3,\cdots$. Then, ...
2
votes
1answer
66 views

How to prove this inequality? (proof of limit)

How can I prove that for every e that : $|(5+n^2)^{1/n} - 1| < \epsilon$ ? (considering that n is a natural number). I know by archemidian property that $1/n < \epsilon$, by can this help ...
3
votes
2answers
40 views

How to check whether a given inequality is correct for a large span of integers?

The inequality $\sqrt{n+ 1}−\sqrt n < \frac{1}{\sqrt n}$ is false for all n such that $101 ≤ n ≤ 2000$. Is the statement true?
1
vote
1answer
63 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
8
votes
4answers
336 views

Improving bound on $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$

An old challenge problem I saw asked to prove that $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}} < 3$. A simple calculation shows the actual value seems to be around $2.8$, which is pretty close to $3$ but ...