All Questions
14
questions
-1
votes
1
answer
56
views
Proof that $\sum_{i=0}^{\infty}2^{(-i+1)} < \sum_{i=1}^{\infty}2^{-(i+1)} \cdot F_i $
I am Facing a problem when learning about data compression.
I learned that the Fibonacci code for data compression is complete it means it can be represented as complete node tree.
I am trying to ...
- 69
3
votes
1
answer
168
views
Show $\sum^M_{i=1}a_i/\sqrt{a'_{i-1}}\leq(\sqrt2+1)\sqrt{a'_M}$ for non-negative integers $a_i$ with $a_n\leq a'_{n-1}=\max\{1,\sum^{n-1}_{k=0}a_k\}$
Let $\{ a_0, a_1, \dots\}$ be a sequence of non-negative integers such that for every $n\geq 1$ the sequence satisfies $a_n \leq a'_{n-1} = \max\{ 1, \sum^{n-1}_{k=0}a_k\}$. Show that
$$ \sum^{M}_{i=1}...
- 561
1
vote
0
answers
36
views
How to bound $\sum_{ j=1}^n j^{n-j}?$ [duplicate]
I came across this sum in some research I'm doing. I need to bound
$$S_n=\sum_{j=1}^{n}j^{n-j}.$$
One bound I've managed is obtained by doing the following: Divide by n! and observe
$S_n/n!=\sum_{j=1}...
12
votes
4
answers
562
views
Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$
Problem: Let $x_1, x_2, \cdots, x_n$ ($n\ge 2$) be reals. Find upper bounds for
$$\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}. $$
...
- 40.3k
-2
votes
1
answer
126
views
How can I prove $\sum_{i=1}^n \sum_{j=1}^n \frac{a_ia_j}{i+j-1}$ is never negative for any set of n real numbers $a_i$ [closed]
I can't figure this out, can someone help me prove it? I know you guys will come up with an incredibly elegant solution
- 1
1
vote
3
answers
71
views
Bounding sum by (improper) integral
I am trying to verify the following inequality that I came across while reviewing some analysis exercises online:
$$
\sum_{n=1}^{k} \left(1-\frac{n}{k}\right)n^{-1/7}\leq \int_{0}^{k}\left(1-\frac{x}{...
- 35
4
votes
2
answers
366
views
Improve upper bound for double sum
Let $0<x<1$.
Is it possible to find a reasonably good upper bound (depending explicitly on $x$) for the following expression?
$$T(x)=\sum_{i,j=0}^\infty \frac{\sin(\pi(2i+1)x)}{(2i+1)(2j+1)[(2i+...
- 39
1
vote
0
answers
175
views
Good upper bound for $\sum_{n=1}^N a^{-n} n^{-b}$, for $a, b \in (0, 1]$
Let $a, b \in (0, 1]$ and define $S_N:=\sum_{n=1}^N a^{-n} n^{-b}$.
Question
What is a good upper bound for $S_N$ ?
Observations
By a simple (and probably careless) application of Cauchy-Schwarz, ...
- 9,575
0
votes
0
answers
144
views
Upper and Lower bound of a finite sum
Find upper and lower bound for the following finite sum:
$$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3} $$
My attempt is:
Using the integral test:
we know that $\frac{1}{1} + ...
0
votes
1
answer
1k
views
find the upper and lower bound for a finite sum
Find upper and lower bound for the following finite sum
$1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$
My attempt:
$1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
- 23
1
vote
0
answers
215
views
Infimum of sum of two values.
Let the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R_{\ge 0}}$ be defined as
$$
f_{n}(x)=\sum_{i,j=1}^{n}\frac{(-1)^{i+j}\cos(\ln \frac{i}{j})}{(ij)^{x}}\quad \forall n\in\Bbb N
$$
There is given ...
- 1,382
0
votes
0
answers
42
views
Upper bound on some sum
Let $n\geq 1$, $\epsilon>0$, $v\in \mathbb{R}^n$ and $c>0$. I want to prove that:
$$\sum_{k=1}^n \left|v^2_{k+1}-v^2_k \right|\sqrt{\epsilon \frac{k}{n^{\frac{1}{3}}}+c} \leq \sqrt{\epsilon}n^\...
- 1,408
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
18
votes
2
answers
865
views
Proving $(λ^d + (1-λ^d)e^{(d-1)s})^{\frac{1}{1-d}}\leq\sum\limits_{n=0}^\infty\frac1{n!}λ^{\frac{(d^n-1)d}{d-1}+n}s^ne^{-λs}$
Question
Let $\lambda \in (0,1), s \in (0,\infty), d \in \{2,3,\dots\}$ and show that in this case the following inequality holds:
$$(\lambda^d + (1-\lambda^d) e^{(d-1)s})^{\frac{1}{1-d}} \leq \sum_{...
- 1,135