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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
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0answers
11 views

How to calculate the integral $I=\int_{0}^{1}{e^{arctgx}dx}$

How to calculate the integral $I=\int_{0}^{1}{e^{arctgx}dx}?$ My attempts using the partitioning method or the variable change method did not result in the desired result. Thank you!
0
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1answer
11 views

Proving that a function is Riemann-stieltjes integrable

Let $g$ a increasing function, and $f$ integrable with respect to $g$ in $J=[a,b]$ proof that $|f|$ is integrable with respect to $g$ By definition if $f$ is integrable with respect to $g$, for ...
4
votes
2answers
58 views

Find all roots of the equation :$(1+\frac{ix}n)^n = (1-\frac{ix}n)^n$

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis. I request to help find the solution. ...
2
votes
0answers
17 views

$(Y,d)$ is a complete metric space $\rightarrow$ $(B(X,Y),p)$ is a complete metric space.

$(Y,d)$ is a complete metric space $\rightarrow$ $(B(X,Y),p)$ is a complete metric space. Where $B(X,Y)$ is the space of all bounded functions from $X$ to $Y$ and $p$ is the sup norm given by: $p(...
0
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0answers
10 views

Goodness of discrete approximation

I am given a function $f: R \times R \to R$ and a random variable $Y$ for which all finite moments exist. I want to approximate conditional expectation $\mathbb{E}(f(X,Y)|X=x)$ with a finite sum $\...
-1
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0answers
20 views

How can I calcute an asymptote of integral?

I have $f(x)=\int\limits_0^x g(t)dt$, and I need calculate an asymptote of $f(x)$ for $x\rightarrow\infty$. Do you know some ways to do this? I think that here can be useful Taylor series.
2
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4answers
64 views

The asymptotic behavior of $n\ln n -n$

How do I show that $$\displaystyle\lim_{n\to\infty}\dfrac{n\ln n - n}{\ln n!}=1?$$
0
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0answers
39 views

Prove for $k\in \mathbb{Z}$ that $\lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$

Prove for $k\in \mathbb{Z}$ that $\lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$ My attempt: Observe $k=1$, then $\lim\limits_{n\to\infty}\sqrt[n]{n}=1$. Let $x_n:=\sqrt[n]{n}-1$. Then: $$n=(1+x_n)^n>1+...
0
votes
2answers
27 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 ...
0
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1answer
70 views

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

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})...
2
votes
3answers
62 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
3answers
55 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 ...
1
vote
0answers
17 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 ...
0
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0answers
28 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
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0answers
31 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
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
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0answers
41 views

Find limit of sequence defined by sum of previous terms [on hold]

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 ...
0
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2answers
36 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
0answers
28 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
72 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 ...
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} $, ...
2
votes
0answers
48 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 ...
2
votes
3answers
58 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 ...
2
votes
1answer
28 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
31 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
19 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
vote
1answer
26 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)...
0
votes
0answers
39 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
2answers
37 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 ...
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 ...
-4
votes
0answers
17 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. ...
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
28 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
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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
0answers
24 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
0answers
22 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 $C$ ...
0
votes
1answer
10 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 ...
6
votes
2answers
268 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_{\...
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 $$\...
1
vote
1answer
40 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_{...
0
votes
1answer
40 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=...
-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\...
4
votes
2answers
58 views

Is the sum of this series a differentiable function?

Let $$f(x) = \sum_{n=1}^{\infty} \frac{1}{nx} \left( 1 - \frac{1}{e^{ \frac{x}{n}}} \right) \wedge x>0$$ Is the sum of this series a differentiable function? my idea For examining ...
0
votes
1answer
22 views

Proving that the composition of derivations on an algebra is not a derivation

This is problem 2.4 in An Introduction to Manifolds in Loring W. Tu Repeating the definition in case it is not standard, a derivation is a linear map $D: C^\infty_p \rightarrow \mathbb{R}$ which ...
1
vote
1answer
46 views

Check if $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2-n^2}$ is continuous and differentiable function

Check if $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2-n^2}$ is continuous and differentiable function $$ D = \mathbb{R} \setminus \mathbb{Z}$$ My try $$\frac{1}{x^2-n^2} \text{~~~} 1/n^2$$ so $$\sum_{n=1}...
-4
votes
0answers
31 views

Real Analysis / Uniform Continuity [on hold]

Must a bounded continuous function on R be uniformly continuous? Justify your answer.
5
votes
2answers
40 views

Calculate $\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}$

Calculate $$\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}$$ My try: $$\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}=\lim _{n\rightarrow +\...
-1
votes
1answer
29 views

Integral function that works as an upper bound

Let $\displaystyle f(x,t)=\left|\frac{e^{-t \sqrt{x}}}{\sqrt{x}}\right|$ where $t>0$ I need to find a function $g(x)$ such that $f(x,t) \leq g(x)$ and $\displaystyle \int_1^{\infty} g(x) \ dx$ is ...
0
votes
1answer
16 views

Phase shift between two oscillating solutions of equal period

Given is an oscillating function $y(t)$ of period $T$ with $y(t) \, \ge \, 0 \, \forall \, t \in \Bbb R$. Consider the differential equation $\frac{\partial g}{\partial t}(t) = a f(y(t)) - \gamma g(...
2
votes
1answer
68 views

Argument about what is infinity mathematically [duplicate]

My brother argues that "infinity is a quantity,an infinite number". I argued back saying infinity is a concept, a quality rather than an actual quantity. My brother was saying that $\frac{5}{\infty} \...