How to prove this inequality involving trigamma functions?
While solving a problem I succeeded to reduce it to the following inequality:$$\forall \{a,b,z\in\mathbb R_+,\ a\ne b\}:\quad...
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Show that $$\sum_{k=1}^n{\mspace{-2mu}\frac{\left\lvert\sin{k}\right\rvert}{k}}\sim\frac{2}{\pi}\mspace{-1.5mu}\sum_{k=1}^n{\mspace{-2mu}\frac{1}{\mspace{-1mu}k}}$$ as $n\to\infty$.Alternatively, since...
View ArticleProve that the set of all open intervals with rational endpoints are countable.
Prove that the set of all open intervals with rational endpoints are countable, I do not know exactly what shall I do, Could anyone help me please?
View ArticleTranslation in $L^{\infty}$
Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$...
View ArticleIf $\lambda$ is an eigenvalue of $T$, then $ |\lambda| \leq n \ \text{max}...
The following exercise comes from "Linear Algebra Done Right", Sheldon Axler, 4th Edition, Section 5A, Exercise 16.Suppose $v_1, \ldots, v_n$ is a basis of $V$ and $T \in \mathcal{L}(V)$. Prove that if...
View ArticleTaylor Series expansion for a composition of functions
The Taylor Series expansion for $\frac{1}{1-x}$ is convergent for every real number $-1 < x < 1$.\begin{equation*}\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n} .\end{equation*}Since $0 \leq x^{2}...
View ArticleWhat is limit of (π/2−arctan(x)) to the power of 1/ln(x) as x tends to zero...
Have seen a similar one on here where x tends to infinity,where they first turned it into an ln / ln
View ArticleApproximate piecewise constant function with continuous function
I have a function $f(t)$ that is piecewise constant:$$f(t) = a_i \forall t\in[t_i,t_{i+1})$$with $n$ values $a_0, a_1, ..., a_{n-1}$, and $n+1$ values $t_0, t_1, ..., t_n$.I want to approximate this...
View ArticleAlternatives to solve these ODEs
I am trying to model some physical phenomena using the following ODE : $f'' = 1/f$ with initial conditions $f(0) = 1/10 , f'(0) = 0$. I have solved it nummerically on a computer and also tried to...
View ArticleWhen I was looking at the proof of Fejer's theorem, I encountered a problem...
How did we get the last equation? Why can the summation be converted into a square term?$$\begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx)\\&= \frac1m\sum_{j=-(m-1)}^{m-1} (m -...
View ArticleFind the limit of the sequence ${a_{n}}$, where $a_{n} =...
QuestionFind the limit of the sequence ${a_{n}}$, where $$a_{n} = \prod_{i=1}^{n}\left(1-\frac{1}{\sqrt{i+1}}\right),$$ if exists.Attempt1.) First, I took $\log$ both sides and that yielded$\ln{a_{n}}...
View ArticleConstant of integration of exponential-type function
The Laurent series about $0$ of $1/{z(e^z-1)}$ is$$\frac{1}{z(e^z-1)}=\frac{1}{z^2}-\frac{1}{2z}+\sum_{n=2}^\infty\frac{B_n}{n!}z^{n-2}$$When we integrate this we...
View ArticleWhat happens to $\bigcap_{ n\in\mathbb{N}}G_{n}$ where $G_{n}$ are open,...
For example if working in $(\mathbb{R},ρ)$ where $ρ(x,y)=\lvert e^x-e^y\rvert$ (this space is incomplete), can we still find a sequence $\{G_{n}\}\in\mathcal{P} (\mathbb{R})$ s.t. $\bigcap_{...
View ArticleExercise with locally convex topological vector spaces
I was trying to solve this exercise from Royden:Let X be a locally convex topological vector space, let $Y \subset X$ be a closed subspace and $x_0 \in X-Y$.Prove that there exists a continuous linear...
View ArticleA question on Gamma function
This might be basic but I have difficulty understanding what exactly goes wrong in the following logic:Consider the Gammma function$$\Gamma(z) = \int_0^{\infty} t^{z-1} \, e^{-t}\,dt \quad...
View ArticlePre Clarkson Inequality
I am actually struggling with a seemingly totally trivial inequality.For any non negative $a,b$ and $p > 2$:$a^p + b^p \leq (a+b)^p$.For natural $p$ this is obvious, but what is with the rationals...
View ArticleHow do I calculate the area of the spherical quadrilateral?
I am taking analysis II and in my homework it's about areas and submanifolds. For one task I have:I should calculate the area of the spherical quadrilateral S ⊂ S2bounded by the meridians −π< ϕ1...
View ArticleIsomorphism in Banach Spaces
Let $E$ and $F$ be Banach spaces. Let $T: E \rightarrow F$ be an isomorphism (i.e., a continuous vector space isomorphism with a continuous inverse). Let $J_E$ and $J_F$ be the canonical injections of...
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Parameterizing Contour Integral
I am trying to parameterize the following integral$$F(\lambda)=\int_{C j} e^{\lambda g(z)} h(z) d z$$where $g(z)=z-z^3/3$ and $h(z)=z^{\mu-1}$ with the curve$$\gamma(t)= \begin{cases}-t-1+i...
View ArticleIs $\lim_{h\to0} \frac{f(x+h(x-y))-f(x)}{h} \geq f'(x)(x-y)$?
Is $\lim_{h\to0} \frac{f(x+h(x-y))-f(x)}{h} \geq f'(x)(x-y)$ for a differentiatable function $f$?I came across this problem and in the accepted solution, the author follows that the above is true, but...
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