All Questions
6
questions
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
1
vote
1
answer
140
views
Other ways to show $y_n = \prod_{k=1}^n \left(1+{1\over x_k}\right)$ is bounded, where $x_n$ is another sequence with given properties.
Problem statement:
Let $n\in \mathbb N$ and $\{x_n\}$ be a sequence of natural numbers such that $S_n$ defined by:
$$
S_n = {1\over x_1} + {1\over x_2} + {1\over x_2} + \dots + {1\over x_n}
$$
...
- 5,411
1
vote
1
answer
96
views
Proof verification of $\sum_{k=1}^n kq^{n-k}$ is unbounded for $n \in \mathbb N$ and $q\in\{\mathbb R\setminus0\}$
As opposed to the sum in this question I want to prove the following:
Given $n\in \mathbb N$, $q\in \{\mathbb R \setminus0\}$ and:
$$
x_n = \sum_{k=1}^n kq^{n-k}
$$
Show that $x_n$ is an ...
- 5,411
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
6
votes
1
answer
1k
views
Upper bound of the sum $\sum_{i=2}^{N}{\frac{1}{\log(i)}}$
One of the questions in Sierpinski's book on number theory lead to finding a tight upper bound for the following sum:
$$\sum_{i=2}^N {\frac{1}{\log(i)}}$$
The trivial upper bound like $\frac{N-1}{\...
- 217
4
votes
1
answer
426
views
Find an asymptotically tight bound for $\sum_{k=1}^nk^r\log^s(k)$
I have an exam coming up that I'm studying for and I'm somewhat stumped by a couple problems. I need to derive an asymptotically tight bound for the following summation:
Assume $r$ and $s$ are ...
- 129