Questions tagged [upper-lower-bounds]

Question 1

Conjecture. Let $(a_i(j))_{j \geq 0}$ be sequences of natural numbers $\geq 1$. For example $a_1 = \overline{2} = 2,2,2,2, \dots$, is the constant $2$, but $a_3 = \overline{2,1,2}$ is not. Define $B =...

Question 2

I should calculate the series: $$\sum_{n=1}^{\infty} \left[ 2^{\frac{1}{n^3}} - 1 - \frac{1}{\sqrt{n}} \right]$$ Consider the sequence $$ a_n = 2^{\frac{1}{n^3}} - 1 - \frac{1}{\sqrt{n}}. $$ Let us ...

Question 3

The $n$th degree Taylor polynomial at $x = a$ is: $$P_n(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n $$ As $n$ gets larger, the Taylor polynomial approximates a function $...

Question 4

Let $W=\{w_k: 1\le k\le N\}$ be sequence of nonzero, distinct, real numbers with $\sum\limits_{k\ge 1}\frac{1}{|w_k|}<\infty$ and $\xi_0$ be a fixed number in $(0,1)$. Find the uniform bound of $$...

Question 5

Suppose that $\tau>0$ and $r>0$ are real parameters, and define $\alpha:\mathbb{R}^{+}\mapsto\mathbb{R}$ to be $$ \alpha(\omega):=\frac {[(\omega^2+r)\cos^2(\omega\tau)+\frac{\omega}{2}(1-r)\sin(...

Question 6

Let $a_n=4n+1-2 \displaystyle \left \lfloor \frac{n}{2} \right \rfloor$ , for $n \in \mathbb{N}$ i.e : integers $\ge1$ that are odd and not divisble by $3 \quad (\star)$ $a_0=4\times 0+1-2\...

Question 7

Let $P$ be a (simple, convex) polytope in $\mathbb{R}^n$ with at least $n + 1$ vertices. Let $f_i$ be the number of elements with $i$ dimensions in $P$. What is the maximum value that $$f(P) = \frac{...

Question 8

Recently I was learning to evaluate the improper integral $$ I=\int_{-\infty}^\infty\frac{du}{u^2+2} $$ My instructor said that we could write $$ I=\lim_{t\to\infty}\int_{-t}^t \frac{du}{u^2+2}=\lim_{...

Question 9

Specifically its Exercise 2.1 in Boucheron, Concentration Inequalities Let $MZ$ be a median of the square-integrable random variable $Z$ (i.e. $P(Z\geq MZ) \geq 1/2 \text{ and } P(Z \leq MZ) \geq 1/2 $...

Question 10

Recently, I came across a problem that has stumped me: Problem Prove that for some natural number $N$, there are exactly 1000 perfect squares strictly between the consecutive cubes $N^3$ and $(N+1)^3$...

Question 11

I have the function $$g(\theta) = \frac{1}{2 \pi} \int_0^{\infty} \frac{1}{c}\text{exp}\left(-\frac{\theta^2}{4}c\right) \text{exp}\left( -\frac{1}{c}\right) dc$$ and I want to prove that as $|\theta| ...

Question 12

Beck (2017): for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is twice-differentiable, for a given $L>0$ $\beta$-smoothness with respect to the $L_p$ norm for $p \in [1,\infty)$ is ...

Question 13

Cramer, along with his conjecture $$g_n=O(\log^2 p_n)$$ also proved, assuming Riemann Hypothesis, $$g_n=O(\sqrt{p_n}\log p_n)$$ However no explicit estimates were provided. Have anyone made it ...

Question 14

Consider a unit $N$-dimensional hypersphere defined by $$ \left\{ x^Tx=1 \right\} $$ where $x$ represents the standard coordinate vector of a point on the N-dimensional hypersphere. Let $A_N$ be its ...

Question 15

Let $A$ be a set of $n_1$ weighted elements $e_1,e_2,\ldots,e_{n_1}$, where weight of element $e_i$ is $w(e_i)\geq 0$ and let $W=\sum_{i=1}^{n_1} w(e_i)$. Given some parameter $\epsilon>0$, let $S\...

Stack Exchange Network

Questions tagged [upper-lower-bounds]

$a_1,\dots,a_n$ periodic sequences summing to $p_1,\dots,p_n$ over each of their resp. periods, then their sums synch. to some value $\leq\sum_i p_i$.

Approach to solve $\sum_{n=1}^{\infty} \left[ 2^{\frac{1}{n^3}} - 1 - \frac{1}{\sqrt{n}} \right]$

Why does the Lagrange error bound of a Taylor polynomial only use the ($n+1$)-th derivative?

Find a uniform lower bound

lower bound for a messy rational function

A simpler upper bound of $\sum _{ \quad k \le a_n \\ \gcd(k,6)=1} \frac{1}{k}$?

Upper bound on the number of facets of a polytope

On the rigor of improper integrals

Can I have some help on the Probability (specifically bounding random variables) problem? [duplicate]

Exactly $1000$ perfect squares between two consecutive cubes

Prove lower bound of complicated function to show divergence at 0

Finding a upper bound on $||\nabla^2 f(x)||_{p,q}$ for p,q $\neq 2$ that is faster to calculate than eigenvalues or smaller.

Explicit bound on prime gaps assuming RH (Cramer's Theorem)

Hypersphere Surface Area Fraction Upper Bound

Sample complexity of distinguishing instances using weighted sampling

Hot Network Questions