(Dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum\limits_{cyc}\frac1{\sqrt...
An open problem from Art of Problem Solving (AoPS):If $a,b,c,d$ are positive real numbers such that$$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove$$\frac1{\sqrt...
View ArticleApplicability of the method of stationary phase
The stationary phase method deals with integrals such as $$I=\int_a^bf(t)e^{i\lambda g(t)}dt;$$ if there is a stationary point $c\in (a,b)$ then we can say that the above integral is approximately...
View ArticleProving that $m^*(E) = 0.$
$\def\R{{\mathbb R}}\def\N{{\mathbb N}}$Let $E\subseteq \R.$ Assume that for any $x\in E,$ there exists $\delta_x > 0$ so that $$m^*(E\cap(x-\delta_x,x+\delta_x))=0.$$ Prove that $m^*(E) =...
View ArticleDerivative of a Gamma function
To prove $$\Gamma '(x) = \int_0^\infty e^{-t} t^{x-1} \ln t \> dt \quad \quad x>0$$I.e. why can we put the derivative inside the integral? We have$$\frac{\Gamma(x+h)-\Gamma(x)}{h}=\int_0^\infty...
View ArticleIs there a non-formal definition of a polynomial?
A polynomial function in one variable is a function $f$ such that $$f(x) = \sum_n a_nx^n.$$ This definition is purely formal in that it gives a form for a polynomial function, but no characteristics....
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Prove that the following series in a Hilbert space converges.
I need a HINT in the following problem :I am trying to understand why if you have an arbitrary Hilbert Space and any Hilbert Basis then, the sum of the elements in the basis weighted with some square...
View ArticleBoundedness in $L^p$ and convergence in $L^1$ implies weak convergence in $L^p$
Let $p\in(1,\infty)$ and $f_n$ be a sequence in $\mathcal{L}^p(E)\cap\mathcal{L}^1(E)$ such that $||f_n||_p\leq1$ for every $n$ and there exist $f\in\mathcal{L}^1(E)$ such that $f_n\to f$ in...
View ArticleHow to solve this limit: $\lim\limits_{x \rightarrow 0}{\frac{1}{x}-\frac{\ln...
$$ \lim_{x \rightarrow 0}{\frac{1}{x}-\frac{\ln (x+1)}{x^2}}$$Can anyone help me by pointing out where I am wrong while solving this limit? The following is my attempt.$$ \begin{aligned}& =\lim_{x...
View Articleevaluation of an infinite sum of ratios of two factorials [closed]
the integration of the differential equation verified by the associated generating series provides the solution but on condition of canceling the integration constantThe image provided is of the...
View ArticleApplying l'Hôpital's rule to a limit defining a derivative
Today I was given this limit to solve:$$\lim_{x\to a}\frac{x^n-a^n}{x-a}\tag1$$I used l'Hôpital's rule and got $nx^{n-1}$. However, I was pointed out that I cannot use l'Hôpital's rule here, since in...
View ArticleUnder what conditions is there a continuous surjection from $\mathbb R$ to...
Consider a space of smooth real-valued functions on some domain.For instance, have the domain be $X = \mathbb R$ or perhaps some compact domain such as $X = [0, 1]$.For the resulting smooth function...
View ArticleSimplest proof of Taylor's theorem
I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem.By which I mean this: there are plenty of proofs that introduce some arbitrary construct: no...
View Article$\nu = \nu_a + \nu_s, \nu_a \ll \mu \implies \nu_s \perp \mu$
Let $\mu$ and $\nu$ be positive measure. Can we claim that $\nu = \nu_a + \nu_s, \nu_a \ll \mu \implies \nu_s \perp \mu$?When we prove the uniqueness part of Radon-Nikodym theorem, $$ \nu_a + \nu_s =...
View ArticleFallacy in the proof of existence of a one-to-one correspondence from...
Denote by $^{S}F$ the set of all functions from set $S$ to set $F$. For example, let $S=\{0, 1\}$ and $F=\mathbb{N}$, then $^{S}F$ denotes the set of all functions from $\{0, 1\}$ to $\mathbb{N}$. This...
View ArticleYoung's inequality for convolutions in $l^2$
Suppose $a_n$ is a sequence of complex numbers such that $|a_n|$ is in $l^2$. Let$$c_m = \sum_{k=0}^m a_k a_{m-k}.$$I wonder if the following inequality holds$$\sum_{m=0}^\infty |c_m|^2 \leq...
View ArticleCan we find a function $f$ such that $f\in C^\infty(\mathbb{R})\cap...
Can we find a specific function $f$ such that $f\in C^\infty(\mathbb{R})\cap H^1(\mathbb{R})\cap L^1(\mathbb{R})$ but $(\sqrt{f})' \notin L^2(\mathbb{R})$?Here, $H^1(\mathbb{R})$ denotes the standard...
View ArticleIs the sequence $a_n = \sum_{k=0}^n \frac{1}{k!} \int_0^{n-k+1} x^k e^{-x}...
Let$$a_n = \sum_{k=0}^n \frac{1}{k!} \int_0^{n-k+1} x^k e^{-x} dx$$for all $n \ge 0$.Is the sequence $(a_n)_{n \ge 0}$ bounded?The integral $\int_0^{n-k+1} x^k e^{-x} dx$ is the lower incomplete gamma...
View ArticleShow that the pointwise limit of monotone increasing harmonic functions is...
Let $\Omega \subseteq \mathbb{R}^n$ be open and consider a sequence $\{f_k\}_{k \in \mathbb{N}}$, $f \in C^2(\Omega)$ of harmonic functions in $\Omega$ such that $0 \le f_k \le f_{k+1}$ and for which...
View ArticleConvergence about $|n\sin (n^k)|$ [closed]
For which $k \in \mathbb{N} $ does the sequence $|n\sin (n^k)|$ converge when $n$ tends to infinity?
View ArticleContinuity function bounded on half-line
I need to prove the following theorem:Let $f: \mathbb{R} \to Y$, where $Y$ is a normed space, be an additive function. Suppose there exists an interval $(a, +\infty) \subset \mathbb{R}$ such that there...
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