Show that $\int_a^b(f+g) = \int_a^bf + \int_a^bg.$ How should I proceed?
Results that have been proven:Let $f$ be continuous on $[a,b]$.If $\mathcal{P}$ is any partition of $[a,b]$, then $l(f, \mathcal{P}) \leq u(f, \mathcal{P})$.If $\mathcal{P}$ and $\mathcal{Q}$ are any...
View ArticleFind appropriate bound of an integral
Let $t \in (t_0, + \infty)$ for $t_0$ an arbitrarily large fixed constant, $\alpha \in (0,1)$ a number close to 1 and $s \in [3t/4,t]$. I'm wondering if it's possible to prove the following...
View ArticleStrong and pointwise convergence in $L^p$ imply the sequence is bounded by a...
Say that $u_n\to u$ in $L^{p}(\Omega)$ for some $p>1$ and also $u_n\to u$ pointwise a.e. on $\Omega$.How can we prove that there is a function $f\in L^{p}(\Omega)$ such that:$$|u_n|\leq f$$a.e. on...
View ArticleShow that $\exp(D_s)$ converges strongly on $L^2$ to $T_1$ as $s \to 0$.
Consider $L^2 = L^2(\mathbb{R})$. Given any real number $s$, define $T : L^2 \to L^2$ by setting$$(T_s x)(t) = x(t+s) \quad \text{for every} \ x \in L^2 \ \text{and} \ t \in \mathbb{R}.$$For $s \neq...
View ArticleDoes any convergent sequence of unit vectors converge to a unit vector?
Let $(V,\Vert\cdot\Vert)$ be a normed vector space and suppose $\{u_n\}_{n=1}^\infty$ is a sequence in $V$ that converges to $u$. If each term in $\{u_n\}_{n=1}^\infty$ has unit length, can we conclude...
View ArticleProof review for discontinuity of Dirichlet function [closed]
Let f(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}I know how to prove that this function is discontinuous everywhere using epsilon delta, but my goal is to find...
View ArticleIs there a Log-Sobolev inequality for Lebesgue measure on $[0,1]$ on compact...
After searching on the internet for long enough, I would like to pose the question here. I hope there is no duplicate (if there is please let me know)Is it true that, there is a universal constant...
View ArticleShow existence of improper limit
Let be $f:[a,b[~\!\to[A,\infty[$ a bijective and strictly monotonically increasing function and $f^{-1}:[A,\infty[~\!\to[a,b[$ its inverse. We assume that $\lim\limits_{x\nearrow b}f(x)=\infty$. Show...
View ArticleEvaluate $\int_0^{\pi/2}...
The following problem is proposed by Cornel Ioan Valean, that is, to prove that\begin{align}& \int_0^{\pi/2} \operatorname{Li}_{2}^{2}\left(\sin^{2}\left(x\right)\right){\rm d}x=\int_0^{\pi/2}...
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Inequality involving entropies: $\left \|p -\frac{1}{n} e \right...
For a given probability vector $p=(p_1,\dots,p_n)$ with $p_1,\dots,p_n > 0, \sum_{i=1}^n p_i=1$, I want to prove the following inequality$$\left \|p -\frac{1}{n} e \right \|^2_2=\sum_{i=1}^n \left...
View ArticleIs the set of $\varepsilon$-discontinuities closed if it is defined without...
Let $f_n \in C([0,1])$ be a sequence of functions with $f_n(x) \to f(x)$ for all $x$. Consider the set of discontiuities of $f$, which I denote by $\mathcal{D}$. Then, we can define:$$\mathcal{D} =...
View ArticleDoes $f \in L^1 \implies f \log f \in L^1$?
Let $\Omega$ be a bounded domain in $R^3$. Suppose $f \in L^1(\Omega)$ with $f>0$ a.e. Prove or disprove: $f \log f \in L^1(\Omega)$.Attempt: I am stuck at this for a while and unable to make any...
View ArticleA sum of integrals representation involving $\zeta(4)$
Do we have the following $\zeta(4)$ representation included in the mathematical literature?$$\zeta(4)$$$$=\frac{32}{45}\int_0^1 x \arctan^2\left(\frac{1}{x}\right) \operatorname{arctanh}^3(x)...
View ArticleSeries with numbers $\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$
I'm looking for series alike in the literature containing in their summands numbers of the type $\displaystyle Q_n=\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$. The following three series have...
View ArticleFind an example of a discontinuous positive semi-definite real function
Can someone give an example of a discontinuous, positive semi-definite real function $f:\mathbb{R}\to\mathbb{R}$?It is a well known fact that $f(t)= e^{-|t|}$ is a positive semi-definite real function....
View ArticleIf $n, z, x\in \mathbb{R} $ with $n > 1$ and $z
If $n, z, x\in \mathbb{R} $ with $n > 1$ and $z<0$, what values of n exist for$z^{1/n}=x$?
View ArticleProving that a recursive sequence is bounded and monotone
Let $a_0=2\sqrt3$ and $b_0=3$, and define two sequences recursively by $$a_n=\frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}}\quad\text{and}\quad b_n=\sqrt{a_nb_{n-1}}.$$This is an exercise in Jay Cummings...
View ArticleSeries $\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$
I'm looking for series alike in the literature containing in their summands numbers of the type $\displaystyle Q_n=\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$. The following three series have...
View ArticleReal Analysis, series and sequences
For a convergent series of non-negative terms, is the sequence of partial sums upper bounded by the value of the series? By definition of convergence of series, can I immediately conclude such a...
View ArticleUsing analytical concepts to solve the algebraic problem.
Suppose $n\ge 2$. Consider the polynomial $[Q_n(x) = 1-x^n - (1-x)^n .]$Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.I have solve the given problem by showing that...
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