Proof Derivative with Definition [closed]
Given the function $f(x) = |x - a| + |x - b|$, show that the function $f$ is not differentiable at $\{a, b\}$.
View ArticleConfusion About Integral Involving Periodic Functions
Let $f_p(t)$ be a periodic function with period $T$.An identity that we often use is that$$\int_{-\infty}^{\infty} dt \, f_p(t) \left[ g(t) - g(t - T) \right]=0\, ,$$which can be shown by shifting the...
View ArticleQuasi-probability density functions and vanishing expectation
I am interested in the properties of smooth quasi-probability density functions. In two dimensions, these are smooth functions $\rho(x,y):\, \mathbb{R}^2\to \mathbb{R}$ that satisfy the...
View ArticleWhen is the ratio of largest number and smallest number when the sum and sum...
Given $n$ positive real numbers $a_1\geq a_2,\cdots\geq a_n>0$. Assume $\sum_{i=1}^n a_i=b_1, \sum_{i=1}^n a_i^2=b_2$. I want to find an upper bound on $\frac{a_1}{a_n}$, and the condition when this...
View ArticleThe series $\sum_{n=1}^{\infty} (\frac{n!}{3.5.7\cdots (2n+1)})^2 4^n? $...
How to check convergence of the series $$\sum_{n=1}^{\infty} \left(\frac{n!}{3\cdot5\cdot7\cdots (2n+1)}\right)^2 4^n? $$I tried by Ratio test which fails as $\frac{a_{n+1}}{a_n}\to 1$. Thank you.
View ArticleShow $\Delta x\Vert D_x^c u\Vert\leq\Vert u\Vert$
Let $\Delta x>0, N>0$ and $I=\mathbb{Z}/N\mathbb{Z}$. For $u=(u_i)_{i\in I}$, one defines$$(D_x^c u)_i=\frac{u_{i+1}-u_{i-1}}{2\Delta x}\textrm{ for all }i\in I\tag{Def}$$and for...
View ArticleWhy can completeness on $\mathbb{R}$ be defined in terms of lower bounds as...
The original exercise was this: Show that the Completeness Property for the real number system could equally well have been defined by the statement, “Any nonempty set of reals that has a lower bound...
View ArticleIs $L^2(\Omega) =L^\infty(\Omega)$ for bounded domains?
Is it true that a function $f$ on a bounded domain $\Omega\subset\mathbb{R}^n$ is $L^2$ if and only if it is $L^\infty$ on that domain?Suppose $f\in L^2$then $f^2$ is integrable, so it is finite on...
View ArticleProve that $F(x,y) = f(x) - f(y)$ for Lebesgue measurable $f$ on $[0,1]$ and...
I am having trouble with an exercise within the book "Modern Real Analysis" by Ziemer.The exercise appears in chapter 6 (on integration) within section 6.8 on the dual space of $L^{p}$. The exercise...
View ArticleDisprove an altered limit convergence statement.
I found this exercise in a textbook I'm working through:Disprove the following by means of a counterexample:($\forall x$ s.t. $0<|x-a|<\delta, |f(x)-L|<\epsilon$) implies ($\forall x$ s.t....
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The inverse map of the stereographic projection
Let $\mathbb S^1$ the unit circle centered at the origin and the pole $p=(0,1)$. The stereographic projection is the homeomorphism $\varphi:\mathbb S^1\setminus \{p\}\to \mathbb R^1$. In order to find...
View ArticleHow to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and...
How to calculate$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$$and$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$$by means of real methods?This question was suggested by Cornel the author of the book,...
View ArticleProve that $l^\infty$ is complete, The discussion on convergence.
Consider the normed space$(l^\infty, \|\cdot\|) = \{u = \{u_n\}{n \in \mathbb{N}} \subset \mathbb{C} \text{ such that } \|u\| := \sup{n \in \mathbb{N}} |u_n|\}.$I attempted to prove that $(l^\infty,...
View ArticleContinuous function property and right control
Assume $X$ is a compact metric space and $C$ is closed in $X$, and $U$ is an compact neighborhood in $X$ containing $C$. Assume $f:U\times [0,1]\rightarrow X$ is continuous and $f(c,t)=c$ for all $c\in...
View ArticleSeek the minimum of $\frac{a_1}{a_n}$ if $a_1 \ge \cdots \ge a_n > 0$ and...
Problem. Let $n\in \mathbb{N}_{n\ge 3}$. Let $b_1, b_2$ be given real numbers such that $b_2 < b_1^2 < nb_2$. Let $a_1 \ge a_2 \ge \cdots \ge a_n > 0$ be real numbers such that $a_1 + a_2 +...
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Proving the "product" of an absolutely convergent series with a convergent...
Hi, in the question shown in the photo, I think I have come up with a proof for part a). However, reading through parts b) and c) I realized that the convergence of the c-series is only guaranteed if...
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Real analysis - can someone please show me how they got this inequality?
I really don't get how they got the first inequality:$$\frac{n+1}{3(n+1)^2+1}\frac{3n^2+1}{n}\leq \frac{3n^2+1}{3n(n+1)}.$$How did they come up with that upper bound? Am I missing a general inequality...
View ArticleHow to find the $\lim_{n \to \infty}\frac{n+n^{1/2}+n^{1/3}+\cdots +...
It is true that the limit $\lim_{n \to \infty}\frac{n+n^{1/2}+n^{1/3}+\cdots + n^{1/n}}{n}$ exists, but I don't know what it is.By the calculation results, I guess the limit is 2.
View ArticleTricky application of dominated converegence
Let $a>b>0$, $g\in L^{1}([0,1])$, and let$$f(x):=x^a \int_{0}^{1}e^{-x^b y} g(y)dy,\quad x>0.$$Claim: $\lim_{x\rightarrow +\infty} f(x)=0$ provided that$y^{-a/b} g(y) \in...
View ArticleHolders Inequality
When proving Holder's inequality, why do we assume $\|x_i\|_p=1=\|y_i\|_q$, where $\frac{1}{p}+ \frac{1}{p}=1$.$\|x_i\|_p=(\sum x_i^p)^{\frac{1}{p}}$.Can it not be proven if the equality to unity...
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