Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
6answers
76 views

Limit of $n^2 (\sqrt[n]{n+1}-\sqrt[n]{n})$

Please how can I calculate limit of $n^2 (\sqrt[n]{n+1}-\sqrt[n]{n})$ without using Taylor's expansion.
1
vote
1answer
41 views

Limit of sequence $\lim \limits_{n \to \infty\ }(\ln n)^{1+\frac{(\ln(\ln n))^{2019}}{\ln n}}-\frac{\sqrt[2018]n}{\ln n}$

$\lim \limits_{n \to \infty\ }(\ln n)^{1+\frac{(\ln(\ln n))^{2019}}{\ln n}}-\frac{\sqrt[2018]n}{\ln n}$ I can show that $\lim \limits_{n \to \infty\ } {\frac{(\ln(\ln n))^{2019}}{\ln n}}=0 $, so $\...
0
votes
4answers
68 views

Limit of $\frac{2^n}{n!}$ [duplicate]

How can I prove that $\lim_{n\to\infty} \frac{2^n}{n!}=0$?
2
votes
3answers
26 views

Comparing integral and series of $1/(x^a)$

The problem is to $$\sum^N_{n=2}\frac{1}{n^a}\leq\int^N_1\frac{1}{x^a}$$ and to use this to prove the convergence of the series for $a>1$. So, I believe I have the second part down. Namely ...
3
votes
2answers
68 views

Calculate $\lim_{x \to 0} \frac{\ln(1+2x)}{x^2}$ with the help of l'Hospital's and Bernoullie's rule.

Task: Calculate $$\lim_{x \to 0} \frac{\ln(1+2x)}{x^2}$$ with the help of l'Hospital's and Bernoullie's rule. My thoughts: Because $\mathcal{D}(f)=\{x\mid x\in\mathbb{R} \land x\neq 0\}$ the ...
2
votes
2answers
81 views

Is $A=\{f(x)\in\mathbb{R} : ||x||=1\}$ an interval if $f:\mathbb{R}^2\mapsto \mathbb{R}$ is continous?

So I have $f:\mathbb{R}^2\mapsto \mathbb{R}$ a continous function and the set $A=\{f(x)\in\mathbb{R} : ||x||=1\}$ I have to prove that A is an interval, but I don't have any idea on how to do it. What ...
1
vote
2answers
36 views

Showing that $Y = \cup_{n=1}^\infty Y_n$ is dense over the Hilbert $H$.

Source of question : Trying to prove that if the Hilbert $H$ has an orthonormal basis, then it is separable. Elaboration : Let $H$ be a Hilbert space and $\{e_n : n \in \mathbb N\}$ an orthonormal ...
0
votes
2answers
41 views

How can I show that this infinite product is nonzero?

How would you show that $\prod_{k=1}^\infty \cos( 2 \pi/3^k)$ is nonzero? Wolfram approximates it as about $-0.37$, and I have a guess that $$ \Big \vert \prod_{k=1}^\infty \cos( 2 \pi/3^k) \Big \...
0
votes
1answer
30 views

Equivalence of $L^p$ norm in a bounded domain

For $1<p<q<\infty$ and a bounded domain $\Omega \subseteq \mathbb R^2$, is the following true? $$C_1\Vert f \Vert_{L^{p}(\Omega)} \leq \Vert f \Vert_{L^{q}(\Omega)}\leq C_2\Vert f \Vert_{L^{p}...
1
vote
0answers
27 views

$ X \subset R^{2}$ be the subset $X=\{(x,y)|x=0,|y|\leq 1\} \cup \{(x,y)|0<x\leq 1,y=\sin(\frac{1}{x})\}$ is

Let$ X \subset R^{2}$ be the subset $X=\{(x,y)|x=0,|y|\leq 1\} \cup \{(x,y)|0<x\leq 1,y=\sin(\frac{1}{x})\}$ Then , X is a) compact b) connected c) path connected set. Graphically ,I have ...
-1
votes
3answers
59 views

Showing that $\int_{0}^{\infty} u^{-\alpha} \sin(u) \, du >0$ for $0<\alpha<1$

Does anyone know how to show $\int_{0}^{\infty} u^{-\alpha} \sin(u) \, du >0$ for $0<\alpha<1$ (without explicitly having to calculate the exact value)?
0
votes
0answers
25 views

Any sequence in metric space has lim supremum

Does any sequence in any metric space have a limit supremum and a limit infimum? If the sequence is growing, it would be $\infty$ if unbounded or a finite number if bounded. If the sequence is ...
0
votes
0answers
26 views

Weak convergence of weak derivative implies strong convergence

Note that here both $w_n$ and $\phi_n$ are vector functions. The following is given : $ w_n \rightarrow w$ weakly in $ L^q(B; {\mathbb{R}}^M)$ & $\{curl[w]\}_{n=1}^{\infty}$ is precompact in $ ...
0
votes
2answers
27 views

Two questions on ODEs: Maximum interval of positivity of solution to linear ODE; Analytic solution to Riccati ODE.

I have two questions about ordinary differential equations: Consider the first-order linear differential equation $x'(t)=a(t)x(t)+b(t)$, with initial condition $x(t_0)=x_0$. Suppose that $x_0>0$. ...
0
votes
1answer
69 views

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $ f(x)=\frac{\sin(x^{3})}{x}$. Then f is not bounded and not uniformly continuous.

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $ f(x)=\frac{\sin(x^{3})}{x}$. Then which of the following is correct: a)f is not bounded and not uniformly continuous b)f is bounded and not uniformly ...
1
vote
2answers
65 views

Proving that $\lim_{n \to \infty}nx_n=\frac{1}{q}$

Suppose $0<x_1<1$, $x_{n+1}=x_n(1-x_n)$, prove $\lim_{n \to \infty} nx_n=1$. Suppose now $0<x_1<\frac{1}{q}$ and $0<q\leq 1$ and $x_{n+1}=x_n(1-qx_n)$, prove $\lim_{n \to \infty}nx_n=\...
0
votes
2answers
30 views

Proof that $b$ is point of closure of $[a,b)$

Let $E=[a,b)$. I want to exhibit that $\{b\}$ is a point of closure of $E$. Attempt at a proof: My goal is to show that for all $\epsilon>0$ there exists a $y\in E$ such that $|b-y|<\epsilon$. ...
2
votes
1answer
36 views

Describe the region where $f_n(x) = nx(1-x^2)^n$ uniformly converges

$f_n(x) = nx(1-x^2)^n$ Describe the region where $f_n(x)$ uniformly converges Try Note : $f_n(x)$ converges pointwise on $(-\sqrt{2}, \sqrt{2})$. I could only prove that, for any interval $I \...
-2
votes
1answer
35 views

What is the integral of $\log\left(x^{2} + k^{2}\right)$ ?.

Where $\displaystyle k$ is a real number ?. I have tried everything but I stuck when I have to find integral of $k^{2}/\left(x^{2} + k^{2}\right)$.
1
vote
1answer
30 views

Show that if $\inf(A^+)=a>0$ then $a\in A$ and $A=\{za;z\in \mathbb{Z}\}$

$A$ is a set such that $x,y\in A\Rightarrow$ $x-y\in A$, and $A^+$ is a subset of $A$ which contains only its positive elements. I was able to successfully show that if $na\in A$, $n\in\mathbb{Z}, n \...
2
votes
1answer
28 views

confusion about measurability requirements for Lebesgue integral

So, it would appear I have forgotten the basic requirements of Lebesgue integration! Let $(\Omega,\Sigma,\mu)$ be an arbitrary $\sigma$-finite measure space. I'm particularly interested in the real ...
0
votes
2answers
39 views

given $x$ irrational can you find $a,b \in \mathbb{Q}$ such that $a+bx = r$ for all $r \in \mathbb{R}$

given $x$ irrational can you find $a,b \in \mathbb{Q}$ such that $a+bx = r$ for all $r \in \mathbb{R}$. I'm trying to solve this. My attempt consists of choosing $b$ close enough to $bx$ such that $...
0
votes
6answers
82 views

Prove that $\tan\left(\frac{1}{x}\right)$ > $\frac{1}{x}$ in the interval [1,∞) [on hold]

I stumbled upon this while I was solving another question, i.e. the comparison test for proving $\sum_{n=1}^∞$ $\tan\left(\frac{1}{x}\right)$ is a divergent series. I would like a rigorous proof or ...
1
vote
1answer
32 views

Describe the interval where $f_n(x) = \sum_{k=1}^n \frac{x^k}{x^k + 1}$ uniformly converges

Find the interval $f_n(x) = \sum_{k=1}^n \frac{x^k}{x^k + 1}$ uniformly converges. Try First, note that $f_n(x)$ converges pointwise when on $(-1,1)$. I claim, $\forall \epsilon \in (0, 1)$, $f_n(x)...
0
votes
0answers
22 views

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous. Note: here $\beta_1$ is the Lebesgue measure restricted to the Borel $\sigma$-algebra in the real line. Im not sure if ...
-2
votes
0answers
36 views

Is this function Lebesque-Integrable? [on hold]

Let $\alpha$ be a real integer. We define the mapping $f : \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ where $f(x) = \alpha \|x\|.$ Show for which $\alpha$ is $f$ Lebesgue-integrable on $...
1
vote
0answers
27 views

Measure of $C^1$ path in $\mathbb{R}^2$

I started studying multivariable integration and still trying to grasp the conecpt of the measure. I`m doing excersices and I keep getting the feeling im doing something wrong so I hope one of you ...
0
votes
3answers
67 views

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then sequence $x_n$ converges to 2.

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then show that sequence $x_n$ converges to 2. I know this sequence is monotonically increasing. But how to prove it converges to 2? The sequence is ...
2
votes
2answers
138 views

Differential equation with “backwards product rule”.

If we have the following differential equation, ($h,f$ known, $y$ unknown): $$f'(x)y(x) + f(x)y'(x) = h(x)$$ it would be easy, since we could spot the derivative for a product: $$(f(x)y(x))' = h(x)$...
0
votes
1answer
38 views

The limit of arithmetric mean of a properly divergent sequence

What is the limit of $$ S_n := \frac{a_1+\ldots+a_n}{n} $$ for any positive interger $n$, where $a_n \to \infty$ as $n \to \infty$? I am trying to show that for any $\beta$ there exist $N$ such that ...
1
vote
1answer
24 views

Understanding the solution key to a problem which shows that the integral of a sum equals a given value.

Suppose that the domain of convergence of the power series $\sum_{k=0}^{\infty} c_{k}x^{k}$ contains the interval $(-r, r)$. Define $$f(x) = \sum_{k=0}^{\infty} c_{k}x^{k} \hspace{1cm} \text{ &...
2
votes
1answer
43 views

Twice diffrentiable function $f: \mathbb R \to \mathbb R$ such that $f''(x)+e^xf(x)=0, \forall x \in \mathbb R$ [on hold]

Let $f: \mathbb R \to \mathbb R$ be a twice differentiable function such that $f''(x)+e^xf(x)=0, \forall x \in \mathbb R$. Then is it true that $f(x)$ is bounded in $[0,\infty)$ i.e. does there exist $...
1
vote
1answer
27 views

How to imply the vanishing gradient condition in KKT?

In Boyd's Convex Optimisation, the following optimisation problem is considered $$ \begin{align} \min\quad & f_0(x)\\ \text{s.t.}\quad & f_i(x)\le 0,\quad i=1:m,\quad m\in\Bbb Z_{\ge 0}\\ &...
-1
votes
1answer
32 views

$A$ is compact and closed then prove …

I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two ...
0
votes
4answers
114 views

Prove that $\sum_{n=1}^∞\frac{\left(\ln n\right)^3}{n^3}$ is a convergent series by using comparison test [on hold]

I proved by using the integral test that the series is convergent but can't find a way to prove by using the comparison test, which was required.
1
vote
1answer
29 views

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove the following properties.

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove: 1) If $\alpha > 0$, show $\sum_{n = 0}^{\infty} a_{n}x^{n}$ converges if $|x| < 1/\alpha$ and diverges if $|x| > 1/\alpha$ ...
0
votes
1answer
31 views

Formula for analytic functions?

In here (third under double infinite series) they list the following formula. $$\displaystyle \sum_{k=0}^{\infty} \sum_{j=0}^{\infty} a_{k,j} = \sum_{j=0}^{\infty} \sum_{k=0}^{j} a_{k, j-k}$$ Is ...
2
votes
1answer
28 views

Existence of polynomial p such that $|f(x) − p(x^2)| < \epsilon$

Let $f$ be a real valued continuous function on $\left[−1, 1\right]$ such that $f(x) = f(−x)$ for all $x \in \left[−1, 1\right]$. Show that for every $\epsilon > 0$ there is a polynomial $p\...
1
vote
0answers
32 views

show that c is a unique fixed point for f over the whole real line R [on hold]

enter image description here I am pretty lost on this stuff, as my professor had us mainly self teach due to lack of time left in the semester. Any help would be appreciated! Thanks
-1
votes
1answer
32 views

Limit of integrable functions is integrable? [on hold]

If $(g_n)$ integrable functions and $\int g_n\to \int g$. Then $\int g<\infty$?
1
vote
1answer
16 views

prove the metric d on $\mathbb{R}^2$ defined by $d((x_1,x_2),(y_1,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^4}$ satisfy the triangle inequality

I'm a bit stuck with this proof. And also, what's the general strategy I need to bear in mind when proving triangle inequality?
1
vote
1answer
46 views

For each interval $[a,b]$ contained in $I$, sequence $\{f_{n}:[a,b]\rightarrow\mathbb{R}\}$ converges uniformly to $f : [a,b] \rightarrow\mathbb{R}$

$I$ is an open interval Using the following fact to show this: ${\{f_n\}}$ converges pointwise on $I$ to the function $f$, and ${\{f'_n}\}$ converges uniformly on $I$ to the function $g$ Attempt: ...
-1
votes
0answers
35 views

Does this integral exist? [on hold]

Suppose $ F $ is a bounded closed set in $ \mathbb{R}^{m} $ ($ m>1 $) and $ f $ a measurable function on an open neighborhood of $F$. Suppose for each $ x\in F $ there is a set $ E\subset F $ ...
1
vote
1answer
36 views

Any countable set has measurable zero

To demonstrate that ANY countable set has measure zero, is it sufficient to show that the natural numbers have a measure zero? If so, why; and, if not, why not? Thank you :)
2
votes
1answer
27 views

Proving the sequence $f_{n} = \sqrt{x^{2} + 1/n}$ converges uniformly to $f(x) = |x|$ on $(-1, 1)$.

I have the following exercise from my book: For each $n \in \mathbb{N}$ and each $x\in (-1, 1),$ define $$f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$$ and define $f(x) = |x|$. Prove that the ...
3
votes
0answers
41 views

Convergence of a series involving cosine

Let $x \in (0, 2\pi)$. Is the series $\sum_{n=1}^{\infty} \frac{\cos(n^2x)}{n}$ convergent? My guess is: YES and I would like to use Dirichlet test: however I have troubles proving that the partial ...
2
votes
2answers
42 views

Prove the definition of the arcsin(s).

I am given $\arcsin: S \rightarrow (-\pi/2,\pi/2) $ is the inverse function of sin(t) (restricted to [$-\pi/2,\pi/2$]). I'm trying to prove that $\arcsin(s)$= $\int_{0}^{s}1/\sqrt{1-x^2}$ . My ...
-1
votes
2answers
38 views

How this series a_j converges? [on hold]

We have $a_n\geq 0$ and suppose that $\sum_{j=n}^{2n} a_j\leq \frac{1}{\sqrt{n}}$. I dont know how to derive that $\sum a_j$ converges.
0
votes
1answer
28 views

How to show the existence of a root for a specific equation?

There is an interesting question about the solvability of the following equation. Let $a, b, c, d$ be constant numbers. In addition, these constant real numbers satisfy exponent $a >1$, finite ...
1
vote
0answers
26 views

Showing Lemma's Fatou for functions not necessarily not negative.

Let $g$ integrable function on $E$ measurable set. Let $(f_n)$ measurable functions and $|f_n|\leq g$ for all $n$. Show that $\int_{E} \liminf f_n\leq \liminf \int_{E} f_n\leq \limsup \int_{E} f_n\leq ...