Questions tagged [upper-lower-bounds]
For questions about finding upper or lower bounds for functions (discrete or continuous).
2,485 questions
1
vote
0
answers
35
views
Formulas to find out the critical limit of decomposing numbers
I'm doing a proof on Graph theory on a colorability problem of a connected graph. My proof method involves decomposing numbers into the sum of smaller numbers according to specific rules. In ...
- 11
1
vote
1
answer
45
views
$a_1,\dots,a_n$ periodic sequences summing to $p_1,\dots,p_n$ over each of their resp. periods, then their sums synch. to some value $\leq\sum_i p_i$.
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 =...
- 22.7k
0
votes
1
answer
140
views
Approach to solve $\sum_{n=1}^{\infty} \left[ 2^{\frac{1}{n^3}} - 1 - \frac{1}{\sqrt{n}} \right]$
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 ...
- 8,908
2
votes
2
answers
124
views
Why does the Lagrange error bound of a Taylor polynomial only use the ($n+1$)-th derivative?
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 $...
- 23
0
votes
0
answers
61
views
Find a uniform lower bound
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
$$...
- 1
0
votes
0
answers
65
views
lower bound for a messy rational function
Suppose that $\tau>0$ and $r>0$ are real parameters, and define $\alpha:\mathbb{R}^{+}\mapsto\mathbb{R}$ to be
$$
\alpha(\omega):=\frac
{[(\omega^2+r)\cos^2(\omega\tau)+\frac{\omega}{2}(1-r)\sin(...
- 187
1
vote
2
answers
137
views
A simpler upper bound of $\sum _{ \quad k \le a_n \\ \gcd(k,6)=1} \frac{1}{k}$?
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\...
- 638
2
votes
0
answers
45
views
Upper bound on the number of facets of a polytope
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
$$f(P) = \frac{...
- 362
8
votes
1
answer
272
views
On the rigor of improper integrals
Recently I was learning to evaluate the improper integral
$$
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_{...
- 143
0
votes
0
answers
26
views
Can I have some help on the Probability (specifically bounding random variables) problem? [duplicate]
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 $...
10
votes
7
answers
1k
views
Exactly $1000$ perfect squares between two consecutive cubes
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 $(N+1)^3$...
- 507
5
votes
4
answers
969
views
Prove lower bound of complicated function to show divergence at 0
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| ...
- 219
1
vote
1
answer
113
views
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.
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 ...
user1300987
0
votes
1
answer
73
views
Explicit bound on prime gaps assuming RH (Cramer's Theorem)
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 ...
- 13
0
votes
0
answers
22
views
Hypersphere Surface Area Fraction Upper Bound
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 ...
- 406