1
vote
2answers
73 views

prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$

Prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$ I already showed that $a_n$ diverges to infinity like this: I used to the lemma which says that if ...
5
votes
1answer
84 views

An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1 $ , then is it true that $52 < a_{1371} < 65$ ? $ EDIT:-$ I am posing another ...
4
votes
2answers
74 views

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

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

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
0
votes
1answer
48 views

How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?

I would appreciate some help proving the inequality $$\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}.$$ Thanks in advance!
2
votes
1answer
55 views

Proof of $\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$

Is it possible to prove the following inequality? $$\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$$ Thanks
4
votes
1answer
95 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
4
votes
2answers
78 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
4
votes
3answers
86 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
2
votes
0answers
41 views

Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=−y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
5
votes
2answers
62 views

Is this sum $S=\sum_{i,j=1}^n\frac{a_ia_j}{i+j}$ greater than or equal to zero?

Given $$(a_1,a_2,...,a_n)\in\mathbb{R}$$ does this inequality hold? $$S=\sum_{i,j=1}^n\frac{a_ia_j}{i+j}\ge0$$ Thanks.
1
vote
1answer
127 views

How prove concave sequence inequality $\left(\sum_{k=1}^{n}a_{k}\right)^2\ge\frac{3n-c}{4}\sum_{k=1}^{n}a^2_{k}$

let concave sequence $\{a_{n}\}$,such $a)_{n}\ge 0$,and such $$\dfrac{a_{i-1}+a_{i+1}}{2}\le a_{i},i=1,2,\cdots,n-1$$ where $a_{0}=0$. show that $\exists c>0$ such that for every ...
2
votes
3answers
61 views

Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
1
vote
3answers
43 views

Question about sequences

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
2
votes
0answers
60 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
0
votes
1answer
36 views

How to show that the inequality is true?

Let $ (a_n)_{n \geq 1}$ be a sequence in $\mathbb{R}$ and let $a \in \mathbb{R}$. Assume that $N \in \mathbb{N}$, $\epsilon > 0$ and for all $n > N$ the following is true: $|a_n - a| < ...
5
votes
3answers
140 views

Inequality $\sum\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum \frac{1}{x + n^2} $

$x\geq0$, then, we have $$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2} $$ The problem is not easy, even $x=1$. Any help will be appreciated
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 ...
7
votes
2answers
115 views

On the numbers divisible by all the Integers not exceeding their $r^{th}$ roots.

Consider the set of all numbers which are divisible by all natural numbers not exceeding their square root, and denote this set by $S_2=\{1,2,3,4,6,8,12,24\}$ (Here the subscript indicates that we're ...
1
vote
1answer
138 views

Inequality with pi

Prove that, for any sequences of real numbers $\{a_n\}$ and $\{b_n\}$, we have $$\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{a_mb_n}{m+n}\le \pi\left(\sum_{m=1}^\infty ...
2
votes
2answers
54 views

Given two series $ x_n $ and $ y_n $

Let $ x_n = \sum_{k=n}^{\ 2n-1} \frac{1}{k} $ , $ y_n = \sum_{k=n+1}^{\ 2n} \frac{1}{k} $ b) Show that $ y_n \leq \ln2 \leq x_n $ for all $n$
0
votes
1answer
47 views

How prove this $b_{n}>b_{n+1}$, if $a_{1}=1,a_{n+1}=a_{n}+e^{-a_{n}},b_{n}=a_{n}-\ln{n}$

Question: let sequence $$a_{1}=1,a_{n+1}=a_{n}+e^{-a_{n}}$$ let $$b_{n}=a_{n}-\ln{n},n\in Z$$ show that: $$ b_{n}>b_{n+1}$$ where $$\ln{n}=\log{n}$$ my idea: since ...
1
vote
2answers
45 views

Check: Convergence of an infinite series

More a check than a question - I just need to ensure that my logic is correct (I always had trouble with this): Show whether the series $$\sum_{n=110}^{\infty}\frac{1}{3^{n}n^{3}}$$ Is divergent, ...
3
votes
2answers
60 views

An inequality for some series

Consider real positive numbers $t_1,t_2,\cdots, t_n$ for some $n\in\Bbb N$, with $\sum_{i=1}^nt_i^2=n$, such that if $0<t_i<1$ then $$\frac{t_i}{\sin\left(\frac{t_i\pi}{1+t_i}\right)}<1$$ ...
0
votes
1answer
24 views

Finding radius of convergence

I have gotten the problem almost solved, but I'm hung up on how to solve this inequality: $$|x|/|2x+1|<1 $$ I could move the denominator to the right side of the equation: $$|x|<|2x+1|$$ But ...
1
vote
2answers
56 views

Odd series convergence

Prove that we have following inequality: $1+ \frac{1}{3} + \frac{1}{5} + ... + \frac{1}{397} > \frac{9}{4}$ Anybody can help me to figure it out?
6
votes
4answers
286 views

Prove that $\sqrt{n} > \ln n$

Prove that $\sqrt{n} > \ln n$ for all $n \in \mathbb{N}$. I need to use this fact for one of the proofs that I am working on. However, I am having trouble proving this. I tried induction but don't ...
-2
votes
1answer
29 views
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
46 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
193 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
19 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
32 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
122 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
60 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
75 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
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 ...
0
votes
1answer
232 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
71 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
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 ...
1
vote
1answer
55 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
64 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
63 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
19 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
32 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
75 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 ...
2
votes
1answer
53 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 ...
11
votes
3answers
307 views

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

For any positive 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
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 ...
1
vote
0answers
66 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 := ...