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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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0answers
9 views

Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
0
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0answers
25 views

$(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers.

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that $$ (x+y)^r \le x^r+y^r? $$ In all the examples I considered, ...
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1answer
18 views

Conditionally convergent series, true or false

It is given that the series $\sum_{n=1}^{\infty}a_n$ is convergent but not absolutely convergent and $\sum_{n=1}^{\infty}a_n=0$ denote by $S_k$, the partial sum $\sum_{n=0}^{k}a_n,~k=1,2,...$, then $...
0
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0answers
16 views

Exercise about ordinary differential equation

I have some problem about ordinary differential equation. Please help me solve and explain it. Thank you very much. Good health! Problem 1. Suppose $y(x)$ is a solution of the equation $$y''-2y'+y=2e^...
0
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0answers
9 views

mean of a field along the line

What is the proof of a mean field along any line? Or how can we define it? i.e \begin{equation} a=\int_0^1 b \, dx \end{equation} where $a$ is the mean of $b$ and $b(0)=b(1)=0$.
3
votes
2answers
47 views

Show power series converges for every $x$.

Let $$f(x) = 1 + a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...$$ be a solution of the differential equation $f'(x)=xf(x).$ Now I need to explain that the power series that define $f(x)$ converges ...
3
votes
1answer
30 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
0
votes
1answer
40 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
1
vote
2answers
20 views

For some $c \geq 0$ $\text{sup} \ \{ c \cdot f(x): \text{some domain of $x$} \}$ = $c \cdot \text{sup} \ \{f(x): \text{same domain of $x$} \}$ [duplicate]

How would you formally justify this? Or is it just notationally obvious? (As opposed to 'conceptually' obvious, which is never an excuse in mathematics.) Edit: For some $c \geq 0$ $\text{sup} \ \{ c ...
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0answers
11 views

Exercise of total variation of subintervals

I got stuck with the following exercise: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is continuous at $a$. The space $BV(I, E)$ ...
1
vote
0answers
26 views

Can we write the following limit -equality?

We know: 1-) The following condition is valid ONLY with zeros (any) for $\,\,\,\,\,$ $0<x<2$ and $x\in \mathbb R $: $$f(x_o)=g(x_o)=0 \,\,\,\,\ ⇔ \,\,\,\,\ f(2-x_o)=g(2-x_o)=0$$ 2-) ...
1
vote
0answers
42 views

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$. Since the question is asking for a sequence of continuous ...
3
votes
3answers
31 views

$g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$, which of the following option is true?

Suppose $f$ is an increasing real valued function on $[0,\infty)$ with $f(x)>0$ for all $x$ and let $g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$; $0 < x <\infty$, Then which of the following are true:...
0
votes
1answer
29 views

$\liminf_\limits{n\to\infty}1_{A_n}(x)=1$ $\implies$ $\lim_\limits{n\to\infty}1_{A_n}(x)=1$?

Source: Partial proof from textbook: I've omitted the case where $x\in A^c$ as it's not relevant. I've also highlighted the part I'm having trouble with in blue. Here is my attempt at explaining ...
0
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0answers
23 views

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? [on hold]

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? Are they just the same thing?
1
vote
1answer
29 views

$[F]_p\le [f]_1[g]_p$ for $1\le p\le\infty$

For real-valued functions $f$ and $g$ on $(0,\infty)$, let $$F(x)=\int_0^\infty f\left(\frac{x}{y}\right)g(y)\frac{dy}{y}$$ If $1\le p\le\infty$, set $$[h]_p=\left(\int_0^\infty |h(x)|^p\frac{dx}{...
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0answers
24 views

Condition for distinct roots of a polynomial. [on hold]

Find all real c, such that $x^5-5x=c$ has distinct roots.
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0answers
47 views

$\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$

Prove that exist $c>0$ such that $\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$ for all $f$ $\in$ $C^2(0,1)$. Maybe that'll help, we can use similar statement about supremums: $\sup_{(0,1)}|...
0
votes
1answer
34 views

How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
0
votes
1answer
23 views

Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \geq N$} is compact.

Define the set $A \subseteq {\ell}^2$ by $A = ${{$x_n$} $\in {\ell}^2 : \sum_{n=0}^{\infty}(1+n){|x_n|}^2 \leq 1$} i) Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \...
0
votes
0answers
10 views

$\overline{lim}$ the set of cluster points

I am currently reading an article where the author has the following statement. " $\overline{\lim}_{t\to\infty} u_t$ is the set of cluster points of the sequence $u_k \subset U$ which is nonempty ...
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votes
0answers
46 views

$\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$ [on hold]

Let $f \colon [0,\infty)\to\mathbb{R}$ be a strictly decreasing, continuous function. Suppose $\lim_{x\to\infty} f(x) =0$. Prove that $\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$.
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1answer
17 views

Is induction the correct approach here?

Let $a_1, a_2, \dots, a_n$ (real numbers) be such that $$a_1 - a_2/3 + \dots + (-1)^{n - 1}a_n/(2n - 1) = 0.$$ Prove that $$f(x):= a_1\cos(x) + a_2\cos(3x) + \dots + a_n\cos([2n - 1]x) = 0,$$ for ...
0
votes
1answer
48 views

series that diverges

We consider the sequence $$(u_n)_{n\in\mathbb{N}}$$ defined by $0<u_0<1$ and $u_{n+1}=u_n-u_{n}^2$ for all $n\in\mathbb{N}.$ I want to prove that the serie with general term $\ln(\frac{u_{n+1}...
0
votes
0answers
32 views

Is a partial differential equation satisfied after reduction to a subspace?

I have a $n$th-order non-linear partial differential in $m$-real variables $x_1,x_2, \ldots, x_m$. Assume a function $f$ satisfies this differential equation. I denote this by $$D f(x_1, x_2, \ldots, ...
0
votes
1answer
31 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
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0answers
60 views

Let $f_n$ is continuous function from $[0,1]$ to $\mathbb{R}$ for any natural $n$. And for any $x$ from $[0, 1]$ series $\sum_n f_n(x)$ converge. [on hold]

Prove that exist a positive length interval $[a, b]$ in $[0, 1]$ such that partial sums of series $f_n$ is evently limited in $[a, b].$
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votes
1answer
26 views

I need to know if.my example for this problem solved

Let $f(x)=\sup(x^3, x^2+1)$, $I=[1,4]$ $\sup(x^3)=64,\ \inf(x^3)=1$ $\sup(x^2+1)=17,\ \inf(x^2+1)=2$ So $\sup(x^3,x^2+1)=\sup(64,17) =64$ Is this correct, can I have good example.
0
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1answer
22 views

Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...
3
votes
3answers
73 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
3
votes
1answer
39 views

Can one characterize the set of all $A\subseteq\mathbb{R}$ satisfying $2\cdot A\cdot A\subseteq A$ and $A\cdot(\mathbb{R}\backslash A)\subseteq A$?

This question is a spin-off of this question. When trying to solve that question, we came up with the idea of construction functions using sets $A\subseteq \mathbb{R}$ having the properties that $2xy\...
0
votes
1answer
30 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
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1answer
45 views

When is Taylor series a polynomial?

We know that Taylor series is a rational function if and only if its coefficients satisfy the linear recurrence relation. Can we put further conditions on the coefficients so that the Taylor series ...
2
votes
1answer
35 views

A polynomial whose roots form an arithmetic progression

Let $f$ be a fourth degree polynomial whose roots form an arithmetic progression. Prove that $f'$'s roots also form an arithmetic progression. I didn' t make much progress, I just wrote $f(x) =a(x-b-r)...
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vote
2answers
50 views

Prove $f(x)= 1$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \notin \mathbb{Q}$ is not integrable.

I want to prove that $$f(x) = \begin{cases} 1 \text{ for } x \in \mathbb{Q}\\ 0 \text{ for } x \notin \mathbb{Q} \end{cases}$$ is not integrable on $[0,1]$. Now I'm at the point in the book ...
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votes
0answers
34 views

Help calculate the limits [on hold]

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
1
vote
0answers
35 views

Application of Hahn-Banach Theorem?

I am going to quote some versions of Hahn-Banach Theorem and try to deduce a statement which might be wrong. Thm 1.1 Let $E$ be a real normed vector space and $F\subset E$ as subspace. $\lambda:F\to ...
0
votes
0answers
23 views

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. [duplicate]

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. Prove that$\displaystyle\lim_{x\to\infty}\frac{f(x)}{x}=0$. I am trying to use cauchy ...
3
votes
1answer
109 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
1
vote
1answer
32 views

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
0
votes
0answers
24 views

Convergence in measure on a finite and infinite measure space

Let $(E,\mathcal{A},\mu)$ be a measure space. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions, which converges in measure to $f:\forall \epsilon>0,\lim_n\mu(\left\{|f_n-f|>\...
0
votes
2answers
73 views

what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$ Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am ...
-2
votes
3answers
89 views

How do I solve $\int_0^1(x-1)^2(1-x)dx$? [on hold]

How do I solve $\int_0^1(x-1)^2(1-x)dx$?
0
votes
1answer
13 views

Condition upon integral divergence

I have had no classes in analysis and was wondering if the following is true, and if so, how one proves it. Proposition: Let $f$ be a continuous function along the domain $(a,b)$. Let $f(a)$ be ...
-4
votes
2answers
51 views

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ [on hold]

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ We have $-1\leq\cos{x}\leq 1$. So $(\cos{x})^n \to 0$ as $n\to \infty$ Please solve this problem. Please find the sum.
0
votes
1answer
13 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
2
votes
1answer
23 views

For a triangular array, if $\max_{1\le k\le n}x_{n,k}\to 0$ and $\sum _{k=1} ^nx_{n,k}\to \lambda$ does $\prod_{k=1}^n(1+x_{n,k })\to e^{\lambda}$?

Consider the following: we have a triangular array of nonnegative numbers $$x_{1,1 } \\x_{2,1 } \ x_{2,2 } \\ x_{3,1 } \ x_{3,1 } \ x_{3,3 } \\... $$ The maximum on each row converges to zero: $\...
0
votes
0answers
31 views

Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...
1
vote
0answers
17 views

prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
0
votes
0answers
23 views

Best bound for the remainder of two variables Taylor's theorem

Let $f:\mathbb{R}^2 \to\mathbb{R}$ be a two times differentiable function. Then Taylor theorem says that there exists $t\in [0,1]$ such that \begin{align*} f({\bf x})=f({\bf a})+\sum_{i=1}^{2}\frac{\...