Cauchy Product of Summable Sequences is Cesàro Summable
According to Wikipedia, if $a_\bullet,b_\bullet$ are real (or complex) sequences such that $\sum_n a_n\to A$ and $\sum_n b_n\to B$,...
View ArticleShowing a certain type of function on $\mathbb R ^d$ is Lipschitz?
I want to prove the following:Assume that $f:\mathbb R ^d \to \mathbb R$ is continuous, convex and $|f(x)|\leq a+b|x|$. Then $f$ is Lipschitz.I thought it would follow immediately from the assumptions,...
View ArticleSup of a continuous function over a rectangular region
Suppose I have a continuous function $f$ on $K \times \mathbb{R}$ where $K$ is a compact subset of $\mathbb{R}^n$, such that $\sup_{y \in \mathbb{R}} f(x,y)$ is finite for all $x$ (so component wise...
View ArticleRudin's Construction of Lebesgue Measure 2
I am currently studying Rudin's RCA book and I have a question about Theorem 2.20, where the author constructs Lebesgue measure on $\mathbb{R}^k$.Here are the definitions and the notations I am basing...
View ArticleConstruct a sequence $(a_n)_{n=1}^{\infty}$ which has exactly three limit...
Definition. Let us say that a sequence $(a_n)_{n=M}^{\infty}$ of real numbers has $+\infty$ as a limit point iff it has no finite upper bound, and that it has $-\infty$ as a limit point iff it has no...
View ArticleA physicist studying math as a hobby -- any recommendations?
I have a background in theoretical physics, but no rigorous training in math besides the introductory coursework I took in college a while ago. I would like to become more proficient in math, though --...
View ArticleIs $\sum_{j=1}^nja_j=o(n)$ as $n\to\infty$?
Suppose that $a_j$ are non-negative real numbers such that $\sum_{j=1}^\infty a_j<\infty$. Is it true that $$\sum_{j=1}^nja_j=o(n)$$ as $n\to\infty$?I am not sure if it is true or not. It is...
View ArticleApplying fundamental theorem of calculus where upper integration bound for...
Normally, the FTC is stated like this:Fundamental Theorem of Calculus: Let $f$ be Riemann integrable on $[a, b]$. For $x \in$$[a, b]$, define $F(x)=\int_a^x f$. If $f$ is continuous at a point $x...
View ArticleHow to evaluate $\lim_{n \to \infty} \frac{\sum_{k=1}^{n} x_{k} }{n}$ with...
How to evaluate $\lim_{n \to \infty} \frac{\sum_{k=1}^{n} x_{k} }{n}$ with $x_{n+1}=\ln \left | x_{n} \right | ,x_{1}=2$?I'm not sure whether it is an example of ergodic theory or not.
View ArticleAsymptotic expansion of $\int_0^{+\infty} \frac{ne^{-\sqrt{t}}}{1+n^2t^2}\,dt$
I would like to get a general formula to get the asymptotic expansion at the order $n$ (so at whatever precision I want) of the following integral :$$I = \int_0^{+\infty}...
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The definition of locally Lipschitz
Marsden's Elementary Classical Analysis seems to indicate this definition:A function $f:A{\subset}\mathbb R^n\to\mathbb R^m$ is locally Lipschitz iffor each $x_0{\in}A,$ there exist constants...
View ArticleHow to reconcile the existence of the least upper bound?
A set of reals that is bounded above has the least upper bound.It is not intuitively clear to me that it should be true. I am aware of constructions of the reals from rationals where this statement is...
View Article$\mathscr{L}^{p_1}(\mathbb{N},\mathscr{A},\mu)\subseteq\mathscr{L}^{p_2}(\mat...
I need to prove the following:Suppose that $1\leq p_1<p_2<+\infty$. Let $\mu$ is the counting measure on the $\sigma$-algebra $\mathscr{A}$ of all subsets of $\mathbb{N}$. Then...
View ArticleShow that $\frac{x^3+y^3}{x^2+y^3}$ is not differentiable at $(0,0)$
Let be $f:\mathbb{R}^2\to\mathbb{R}$ where$$f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^3},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases}$$Show that $f$ is not differentiable at $(0,0)$.My...
View ArticleSolving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) =...
I need to solve the following equation:$Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and second...
View Article$f(x) \ge 0$ for all $x \in [a, b]$ and $\int_a^b f = 0$. Prove that $f(x) =...
Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.My attempt:Let $\dot{\Pi}$ be a tagged...
View ArticleDo all series have a closed form representation of their partial sum? If not,...
The question was motivated by the way in which we approach the convergence and divergence of some series.During my undergraduate analysis course one of the only times in which the partial sum was used...
View ArticleSome inequalities resulting from the fact that a binomial coefficient divides...
I need to know some inequalities resulting from the fact that a binomial coefficient${n \choose k} $ divides another binomial coefficient ${m \choose j} $. My searching returned no result during...
View ArticleShow that $\frac{x^3+y^3}{x^2+y^2}$ is not differentiable at $(0,0)$
Let be $f:\mathbb{R}^2\to\mathbb{R}$ where$$f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^2},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases}$$Show that $f$ is not differentiable at $(0,0)$.My...
View ArticleHelp with Infimum and Supremum in inequality.
I have a problem let s2 = {x in R : x > 0}. Does s2 have lower bound, upper bound? Does inf(s2) and sup(s2) exist?I understand the that the lower bound is 0 while there is no upper bound. I think I...
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