Prove whether $x_n=\frac{2^n+n^{2^n}}{(-3)^{3n}-3}$ converges or diverges
Q. Prove whether $x_n=\frac{2^n+n^{2^n}}{(-3)^{3n}-3}$ converges or diverges.Could I please get some help on this sequence question? I have been stuck for far too long now, and cannot manage to do this...
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Can this pattern connect trigonometry and repeating decimals?
I have observed a cyclical pattern between the tangent function and repeating decimals.Consider the following function:$f(x)=\tan(\sum_{k=0}^{8} rep(1/x) \cdot 10^{kd})$,where,$x$ is not a multiple of...
View ArticleIntersection of a $C^\infty$ manifold and a sufficiently small ball
Let $K\subset \mathcal{R}^n$ be a compact and convex set with non-empty interior and suppose its boundary $\partial K$ is a $C^\infty$ manifold. Let $y\in \partial K$.I want to argue there exists a...
View Article$\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on...
On wikipedia, it was mentioned that for $\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on $L^{p}$ if it is bounded on $L^{q}$. I wonder what proof of this would be.For any $f...
View ArticleProblem understanding proof about total derivatives
I don't understand some steps in a proof that shows a function $f:U\subset\mathbb{R}^2 \to\mathbb{R}$ is differentiable when its partial derivatives exist and are continuous. The proof fixes a point...
View ArticleConfused if proof is correct $ \limsup s_nt_n = +\infty$
Given$ \limsup s_n = + \infty $ and $ \liminf t_n > 0$To prove $$ \limsup s_nt_n = +\infty$$There exists $s_{n_k}$ such that $ \lim s_{n_k} = +\infty $. Also we have $ \lim t_{n_k} = \liminf t_n...
View ArticleShow that $f(F(x))F'(x)$ is measurable.
This is a equation from Stein-Sharkarchi Real Analysis. Let $F$ be absolutely continuous and increasing on $[a,b]$ with $F(a)=A$ and $F(b)=B$. Suppose $f$ is any measurable function on $[A,B]$.Show...
View ArticleShowing that $\mathcal{A}$ is a $\sigma$-algebra.
Here is the question I am thinking about:Suppose that $\mathcal{A}$ is an algebra and its closed under countable increasing unions.Show that $\mathcal{A}$ is a $\sigma$-algebra.My thoughts:I want to...
View ArticleAre Besov embeddings strict?
Let $B^{\alpha}_p:=B^{\alpha}_{p,\infty}$ be the Besov space of regularity $\alpha<0$ and integrability $p\ge1$. Recall that a distribution $f$ from the dual Schwarz space is in $B^{\alpha}_p$ if...
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Proving path connectedness of a set
Let $K$ be a compact convex set with nonempty interior. Let $y\in \partial K$. Suppose there exists a ball $B_\delta(y)$ such that $\partial B_\delta(y)\cap K$ is connected. I want to argue that...
View ArticleThe L.U.B of a Set that is Bounded Above but has No Greatest Element is a...
I want to prove that if $S$ is a nonempty set of $\mathbb{R}$ that is bounded from above but has no greatest element, then l.u.b. $S$ is a cluster point of S.I know a direct proof might be simpler, but...
View Articleprove the set $Z^{*}=\left\{ x\in X\mid\exists y\in\mathbb{R}:\forall...
I am trying to prove that the following set $Z^{*}$ is measurable giving the following context:Let $(X,S$) be a measure space, for any $Z\subset X$ we will define $S\!\upharpoonright\!Z=\left\{ Z\cap...
View ArticleProduct of a concave function and a decreasing function
Let $f(x)$ and $g(x)$ be two positive continuous functions.Function $f(x)$ is concave at $x_1$ and function $g(x)$ is $<1$ and is decreasing; $g(0)=1$ and $f(0)=0$.Define function $h$ as the product...
View ArticleDifference sets of discrete sets and iterations thereof
For $E\subseteq\mathbb{R}$, let us define the “difference set” of $E$ as:$$\Delta E := \{x-y : (x,y)\in E^2\}$$Furthermore, when $\mathcal{C} \subseteq \mathcal{P}(\mathbb{R})$,...
View ArticleDoes the series $\sum\limits_{n=1}^\infty \frac{e^{inz}}{\bar zn^3+|z|^3}$...
I attempted to use the Weierstrass M-test:Since $z=\bar z$ on all $A\subseteq \mathbb R$, we have, bounding the term from above, $$\left| \frac{e^{inz}}{\bar zn^3+|z|^3}\right |\leq...
View ArticleWhy does the measurable uniformization property imply all sets are measurable?
In The strength of measurability hypotheses, Raisonnier and Stern formulated the following principle:(MUP). For any family $(A_x)_{x\in B}$ of non-empty subsets of$2^\omega$, indexed by the elements of...
View ArticleMunkres' definition of the extended integral
After defining the Riemann integral over bounded subsets of $\mathbb{R}^n$, Munkres'Analysis on Manifolds defines the improper integral as follows:Definition. Let $A$ be an open set in $R^n$; let $f :...
View ArticleProving Unimodality of a high order polynomial
I have a function\begin{align*}Z(t) &= 4(1-\alpha)(3-5\alpha)t^6+12(1-\alpha)^2t^5+8(1-\alpha)(2\alpha-1)t^4\\&\quad...
View ArticleAbout BMO space on smooth bounded domains
Let $\Omega$ be any domain(open and connected) in $\Bbb R^d$.Define the $\text{BMO}(\Omega)$ space as$$\text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty...
View ArticleFind the maximal region in $\mathbb C$ in which the following series...
Find the maximal region in $\mathbb C$ in which the following series $\sum\limits_{n=1}^\infty \frac{(-1)^{3n}n-n^2}{n+4}z^n$ is uniformly convergentI would like some help with this question.The series...
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