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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.

0
votes
2answers
21 views

Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null sequence (a sequence tending to $0$)

Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ and $(w_n)_{n\geq 1}$(a null sequence) be sequences. Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null ...
1
vote
3answers
52 views

Calculate $\lim _{x \rightarrow 0} \frac{\int _{0}^{\sin x} \sqrt{\tan x} dx}{\int _{0}^{\tan x} \sqrt{\sin x} dx}$

Calculate $$ \lim _{x \to 0}\frac{\int_{0}^{\sin\left(x\right)} \,\sqrt{\,\tan\left(t\right)\,}\,\mathrm{d}t} {\int_{0}^{\tan\left(x\right)}\,\sqrt{\,\sin\left(t\right)\,}\, \mathrm{d}t} $$ I have ...
0
votes
1answer
60 views

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$

I want to prove that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ There is a solution of the sum provided in my text book. There the expansion of $(1+\frac{1}{n})^n$ is like below: $(1+\frac{1}{n})...
0
votes
3answers
46 views

Find $S_{n}=\sum_{k=1}^{n}k!(k^2+1)$

$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$ I need to find $S_n$ I started like this: $S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$ How to continue?I tried to give the k values but the terms ...
0
votes
0answers
23 views

functions convergence

$f$ is bounded function defined around $x_0$. For every monotonic sequence $x_n \rightarrow x_0$, $f(x_n)$ is convergent. prove/disprove : all $f(x_n)$ sequences converge to the same limit when $x_n\...
0
votes
0answers
22 views

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a function. And $f_{xx}$ and $f_{yy}$ exist and continuous. Is $f$ a twice differenced function?

Obviously, it is true if $\delta>0$ imply $2f(x+\delta, y+\delta)-f(x+2\delta, y)-f(x, y+2\delta)=o(\delta^2)$. But this result I can't prove too.
1
vote
0answers
14 views

Characterization of measurable functions mapping into banach space via simple functions?

This is a statement I read in Lang's Real and Functional Analysis Chpt VI, Sec 1, M8. M8. A map $f:X\to E$ with $E$ finite dimensional real vector space is measurable if and only if $f$ is a limit ...
2
votes
2answers
921 views

Discreet weighted mean inequality

Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and ${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let $r$ be a real number. Then for $r\ne0$ , we define ${M_{r}}(a,p)={\left({\...
0
votes
1answer
38 views

Let $\{f_k\}_{k=1}^\infty\in L^{loc}_1(\mathbb R^n)$, show $\{f_k^2\}_{k=1}^\infty$ does not converge in $\mathscr{D}^\prime(\mathbb R^n)$

Let $\{f_k\}_{k=1}^\infty\in L^{loc}_1(\mathbb R^n)$ be a sequence of real value functions such that $\mbox{supp}(f_k)\subseteq\{|x|\leq k^{-1}\}$, $\quad \int f_k(x)dx=1$ $\quad for\quad all \quad k=...
2
votes
1answer
54 views

Show that $X_n \to 0$ in probability under given condition.

Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$ Show that $X_n \xrightarrow{P}{ 0}$. Attempt: We have to show: $$\...
0
votes
2answers
35 views

$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$

$$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$$ I got $S=\sum_{k=2}^{n}\frac{1}{(k-2)!}+\frac{1}{(k-1)!}-\frac{1}{k!}-\frac{1}{k!}$ I give k values but not all terms are vanishing.I remain with $\...
0
votes
1answer
17 views

Prove that if $(z_n)_{n\geq 1}$ is a null sequence, then $(|z_n|^q)_{n\geq 1}$ with $\forall q\in \mathbb{Q}:q>0$ is a null sequence and vice versa.

Let $z\in \mathbb{C}, (z_n)_{n\geq 1} \subset \mathbb{C}$ be a sequence. Prove that: $(z_n)_{n\geq 1}$ null sequence $\Longleftrightarrow$ $(|z_n|^q)_{n\geq 1} \quad,\forall q\in \mathbb{Q}:q>0$ ...
-1
votes
0answers
15 views

Find the equivalent of a sequence [duplicate]

let $u_n$ such as $u_0 > 0$ and $u_{n+1}=u_n+1/u_n$ Show that $u_n \sim \sqrt{2n}$
0
votes
0answers
24 views

Find the order of the distribution $\langle u,\phi\rangle=\sum_{k=1}^\infty\partial^k\phi(1/k)$

Find the order of the distribution $u$ in $\mathscr{D}^\prime((0,\infty))$ $$\langle u,\phi\rangle=\sum_{k=1}^\infty\partial^k\phi(1/k)$$ I guess it does not have finite order, but I have no idea ...
0
votes
2answers
68 views

$f(x)=xg(x)$. Does the continuity of $f$ implies that $g$ is also continuous?

Let $f:\mathbb R^{m\times n}\to\mathbb R^m$ be a continuous function. $x\in\mathbb R^{m\times n}$. $g:\mathbb R^{m\times n}\to \mathbb R^n$. $g(x)$ is a column vector, and $f(x)=xg(x)$. Can we ...
0
votes
0answers
37 views

Find limit of sequence defined by sum of previous terms

I have been trying for days to find the limit of this sequence, I'm desperately hoping that someone could help find it. Any help is very much appreciated! Let a and b be natural numbers, with $1\leq ...
1
vote
0answers
43 views

Average Distance Between Zeroes of $\sin(x)+\sin(x\sqrt{2})+\sin(x\sqrt{3})$

QUESTION: What is the average distance between the consecutive real zeroes of the function $$f(x)=\sin(x)+\sin(x\sqrt{2})+\sin(x\sqrt{3})$$ or, more specifically, if $z(x)$ is defined as the number of ...
0
votes
0answers
37 views

Can analytic continuation be applied to real functions? e.g. $f(x) = a\sin x + b\cos x$ in $[0,17]$

I've a Fourier series up to the 3rd harmonic. It looks like this: f(x) = 7.833333335 - 0.327444444*cos(0.349*x) - 0.882182222*cos(0.698*x) + 0.0000355555*cos(1.047*x) - 5.150566667*sin(0.349*x) - 2....
1
vote
0answers
11 views

An topological basis for $X^*$ equipped with weak-star topology.

Let $X$ be a real normed space. Let $B$ the closed unite ball in $X^*$, where the operator norm has been chosen for distance. Does the collection of all sets in the form $\varepsilon B + L^{\perp} $, ...
0
votes
0answers
16 views

To show principal value integral has order 1 as a distribution

To show principal value integral has order 1 as a distribution I have proved that $|p.v\int \frac{\phi(x)}{x}dx|\leq C\displaystyle\sum_{|\alpha|\leq 1}|\partial^\alpha \phi|$ for some constant $C$ ...
2
votes
3answers
55 views

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
1
vote
1answer
26 views

Prove that $\int_{0}^{2} e^{x^2-x}dx \in [2e^{\frac{-1}{4}}, 2e^2]$

Prove that $\int_{0}^{2} e^{x^2-x}dx \in [2e^{\frac{-1}{4}}, 2e^2]$ I can't calculate this because from Wolfram I know that: $$\int_{0}^{2} e^{x^2-x}dx= \frac{\sqrt{\pi } \left(\frac{\text{erfi}}{2}+\...
0
votes
1answer
30 views

Is there an exception to the law of large numbers?

I was reading about the law of large numbers, and under its strong formulation it says that the sample average converges almost surely. That means that it may exist a finite subset with measure $0$ ...
0
votes
1answer
30 views

Rank of differential (Jacobian) is local minima in continuously differentiable functions

Let $f\in C^1(\mathbb{R}^n\to\mathbb{R}^m)$ and let $a \in \mathbb{R}^n$ Prove there exists an open ball $B(a,\epsilon) $ such that for every $x\in B(a,\epsilon)$: $rankD_f(a)\le rankD_f(x)$ ...
12
votes
3answers
787 views

Solving an infinite sum using integration or derivation

There is an infinite sum given: $$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$ It should be solved using integration, derivation or both. I think using power series can help but I don't know how to finish ...
1
vote
1answer
24 views

Find $\lim_{(x, y)\to(1, 1)}\frac{x^y-y(x-1)-1}{(x-1)^2 + (y-1)^2}$

$\lim_{(x, y)\to(1, 1)}\frac{x^y-y(x-1)-1}{(x-1)^2 + (y-1)^2}$ I am trying to use squeeze theorem but I am having trouble with finding an upper bounding convergent to 0 (I believe that 0 is the limit)...
1
vote
1answer
27 views

Change the order of minimum in sums

Is it possible to change the order of minimum as i did below: $$\min_w \sum_x\sum_y f(x,y,w)^2 = \sum_x \min_w \sum_y f(x,y,w)^2 $$ If for all $x$ we have $$\min_w \sum_y f(x,y,w)^2,$$ then could I ...
1
vote
2answers
35 views

Odd Bernoulli numbers are zero using functional equation of Bernoulli Polynomials

Let $(B_k)$ denote the sequence of Bernoulli numbers and let $B_n(x):=\sum_{k=0}^n {n\choose k}B_k x^{n-k} $ denote the Bernoulli polynomials. Bernoulli Polynomials satisfy the following functional ...
3
votes
4answers
242 views

How to evaluate $\lim _{x\to \infty }\left(1+2x\sqrt{x}\right)^{2/\ln x}$?

I have a problem with this limit, can you please show me a way to solve it without L'Hôpital's rule? $$\lim _{x\to \infty}\left(1+2x\sqrt{x}\right)^{\frac{2}{\ln x}}$$ This is my solution (it's ...
0
votes
5answers
92 views

$\lim_{x \to 0} \frac{e^{2x} - \ln(1-x) - \sin(x)}{\cos(x)-1}$ using Taylor Expansions

As a preface- a very similar question is here: Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$ But, my actual question is, when we substitute the ...
2
votes
1answer
87 views

Does the multivariable limit $\lim_{(x,y) \to (0, a)} \frac{\sin(xy)}x$ exist?

With the change of variables, we have $\displaystyle\lim_{(x,y) \to (0, a)} \frac{\sin(xy)}x = \displaystyle\lim_{(x,y) \to (0, 0)} \frac{\sin(x(y+a))}x$ Now we perform the exchange of variables $\...
3
votes
3answers
140 views

Evaluate $\lim_{x \rightarrow 0} \frac{(1-\cos x)^2}{\log (1 + \sin^4x)} $

According to Mathematica, $$\displaystyle\lim_{x \rightarrow 0} \frac{(1-\cos x)^2}{\log (1 + \sin^4x)} = \frac{1}{4}$$ For my purposes it is sufficient to know this limit exists and is finite, but ...
1
vote
2answers
103 views

Evaluate $\lim_{n\to\infty}{ \int_0^{\infty}} \frac{\ln(n+x)}{n}e^{-x} \cos(x)\, dx$

The following is one of my graduate analysis past year paper questions. Question: Evaluate $$\lim_{n\to\infty} \int_0^{\infty} \frac{\ln(n+x)}{n}e^{-x} \cos(x)\, dx.$$ All main steps must be ...
-4
votes
0answers
16 views

Integrals over surface and multiple integral

Which of the integrals $\iint_{S} \sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}}dS$ and $\iiint_{V} \sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}}dxdydz$ is greater and how many times. ...
12
votes
2answers
356 views

Evaluate $\lim_{n\to\infty}nI_n$ with $I_n=\int_0^1\frac{x^n}{x^2+3x+2}dx$

We have to evaluate: $$\lim_{n\to\infty}nI_n$$ with $$I_n=\int_0^1\frac{x^n}{x^2+3x+2}\:dx.$$ There is an elegant way to solve this problem? Here is all my steps: My first ideea was to find a ...
2
votes
1answer
40 views

Prove that limit $\lim_{(x, y)\to(0, 0)}\frac{x^2\cdot y}{x^2 + y^3}$ does not exist. [duplicate]

$$\lim_{(x, y)\to(0, 0)}\frac{x^2\cdot y}{x^2 + y^3}$$ I was trying to find two paths along x-y plane that when followed lead to different limits, but I always got 0.
1
vote
1answer
27 views

Examine the almost uniform convergence

Examine the almost uniform convergence of the function with the formula $$ f_n: \mathbb R \rightarrow \mathbb R \mbox{ such that } f_n(x) = \frac{nx}{e^{(nx-1)^2}} $$. Can somebody check if my ...
-2
votes
2answers
32 views

Find the limit of $a_n=\left(\frac{3n^4+n^2+n(-1)^n}{2n^4+i^n+n^2}\right)_n$

Find the limit of $a_n=\left(\frac{3n^4+n^2+n(-1)^n}{2n^4+i^n+n^2}\right)_n$ I assume that it tends to $1.5$ since I did some tests. For example $a_{10000}\approx 1.5$. I also know that $n^4$ ...
0
votes
1answer
56 views

Double harmonic series $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{H_{n+m}^{(p)}}{(n+1)^{q}(m+1)^{r}}$

Do these sums exist in the literature and have been investigated before? The same question for the odd variant, that is $$ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{O_{n+m}^{(p)}}{(2n+1)^{q}(2m+1)^{...
0
votes
0answers
22 views

real analysis 2 / Rim*

How we can prove that; the line segment has a volume zero using the definition of Jordan region and this is the definition: A subset of Rn is Jordan region if and only if volume for the boundary ...
0
votes
1answer
9 views

Norms in the Space $NBV(\mathbb{R}$)

So the space $NBV(\mathbb{R}$) is the space of bounded variation functions where when $x \rightarrow -\infty$ then $ f(x) \rightarrow 0$. So we defined a norm that is $\| f\|=V_f(\mathbb{R})$. I ...
0
votes
3answers
75 views

Rudin's proof of theorem 1.20b (Archimedean Principle)

The theorem states: (a) If $x\in \mathbb{R}, y\in \mathbb{R}$, and $x>0$, then there is a positive integer $n$ such that $$nx>y.$$ (b) If $x\in \mathbb{R}, y\in \mathbb{R}$, and $x<y$, ...
7
votes
2answers
120 views

Applying the Fourier transform to Maxwell's equations

I have the following Maxwell's equations: $$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$ $$\nabla \...
-1
votes
2answers
24 views

Doubt with definition of divergence sequence to $-\infty$

Good night i have a doubt with definition of sequence diverge to $-\infty$ I know the definition when $\{x_n\}\rightarrow+\infty$ is this: For all $l>0$ exists $N\in\mathbb{N}$ such that is $n\...
0
votes
1answer
23 views

function in $\mathcal{L}^2$ space periodically.

Any idea or hint to prove this theorem ? The book says the hint is by drawing a picture, but I don't really get it. If $f\in\mathcal{L}^2([-\pi,\pi])$, extended periodically in $\mathbb{R}$, then $$\...
3
votes
1answer
116 views
+100

Checking my understanding of the process of developing function into power series

I would like to get some help with the next problem: I'm trying to learn how to develop the real function into the power series. After reading my book, I want to check if I understood correctly what ...
1
vote
1answer
36 views

On the liminf and limsup of the fourier coefficient of non-increasing function

Let $f:(0,\infty)\to\mathbb{R}_+$ be a non-negative non-increasing function such that $\int_0^1 xf(x)\,dx<\infty$. Define $A(b)=\int_0^\infty e^{-x}\sin(bx)f(x)\,dx$. Is it possible that $\limsup_{...
1
vote
1answer
84 views

Brouwer fixed-point theorem on a function

For $f:\mathbb R^n \rightarrow \mathbb R^n \quad \exists R \gt 0 $ that $f$ is continous on $\bar B_R(0):= \{ x \in \mathbb R^n : \Vert x\Vert_2 \leq R \} \subset \mathbb R^n$ . Let also $\langle f(x)...
5
votes
2answers
189 views

Compactness of finite sets

In rudin's analysis books, he defines compactness as: A subset $K$ of a metric space $X$ is ${\bf compact}$ if every open cover of $K$ contains a finite subcover. More explicitly is that if $\{ G_{\...
1
vote
2answers
60 views

Find the minimum using lagrange

I've got the formula $$x^3+y^3+z^3$$ With the constraint $ax+by+cz = 1$ I tried to solve this using lagrange but every possible way I try to use does not get me to the right answer Using the ...