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
5answers
56 views

Collecting proofs for $\sum_{n=2}^{\infty} \, \frac{n-1}{2^n} = 1$

I just stumbled across the fact that $\tag 1 \sum_{n=2}^{\infty} \, \frac{n-1}{2^n} = 1$ This is equivalent to $\tag 2 \sum_{n=1}^{\infty} \, \frac{n}{2^n} = 2$ I discovered $\text{(1)}$ using a '...
1
vote
1answer
37 views

Estimate the remainder term of an unusual interpolation: $f(-1)=f(1)=f'(0)=f''(0)=0$

Suppose that $f(x)\in C^{4}[-1,1]$ and $$f(-1)=f(1)=f'(0)=f''(0)=0$$ Show that for every $x\in[-1,1]$, there exist a $\xi_x\in[-1,1]$, such that $$f(x)=\frac{x^4-1}{4!}f^{(4)}(\xi_x)$$ where $f^{(4)}...
4
votes
0answers
37 views

Exact value of Hausdorff measure of two dimensional Cantor set

Let $\mathcal{C}$ denote the classical Cantor set, then it is well-known that $\mathcal{C}$ has Hausdorff dimension $\alpha = \ln 2 /\ln 3$, and its $\alpha$-dimensional Hausdorff measure is $\mathcal{...
1
vote
2answers
63 views

Prove that the sequence of functions $g_{n}\in C[0,4]$

I want to show that the function defined by $g_n:[0,4]\to \Bbb{R}$, defined by\begin{align} g_n(t)=\begin{cases}0,& \text{if}\;0\leq t\leq 2,\\\dfrac{n}{2}(t-2),& \text{if}\;2\leq t\leq 2+\...
-1
votes
1answer
23 views

Proof about continuity and supremum

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and increasing function. Prove that for all bounded non-empty set of real numbers $A : \sup f[A]=f(sup[A])$.
2
votes
0answers
7 views

Limits of a multiple integral function

Problem Let $f(x)\in L^{1}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$ and $S_t=\left\{x\in\mathbb{R}^N: |x_1|\le t\right\}$ with $t>0$. Let $\phi(t)$ the integral function $$\phi(t)=\int_{\mathbb{...
0
votes
4answers
57 views

Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$

Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$ Let $x = a_{n}/2$. Then $a_{n + 1} = x + 1/x$. Define $f(x) = x + 1/x$ ...
-2
votes
0answers
10 views

Prove that the estimate is correct

How to prove that the estimate is correct: $$ \sum _{n = N+1}^\infty f(n) \leq \int_{N}^\infty f(n) dn.$$ Help me, please.
0
votes
2answers
33 views

Injective mapping

Help me with a simple question, please If $f:X \rightarrow Y$ is a mapping, $f$ is injective if and only if for all set $Z$ and all pair of mappings $g:Z \rightarrow X$ and $h:Z \rightarrow X$, $f \...
4
votes
2answers
34 views

Betweenness preserving implies monotonic?

For this question, we can assume that $f:\mathbb{R}\rightarrow\mathbb{R}$. However, I hope that an answer can generalize to arbitrary linearly ordered sets. I assume that everyone will know what I ...
1
vote
0answers
17 views

Queris realated to “An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$.”

The following proof of An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$ is provided by my professor.I've some ...
3
votes
0answers
60 views

For $A\subset\Bbb R^n$ and $B\subset \Bbb R^m$ show that $\dim_H(A\times B)=\dim_H(A)+\dim_H(B)$

Im stuck with this exercise For $A\subset\Bbb R^n$ and $B\subset \Bbb R^m$ show that $\dim_H(A\times B)=\dim_H(A)+\dim_H(B)$ where $\dim_H$ is the Hausdorff dimension. I know that when $A$ and $B$ ...
3
votes
5answers
63 views

Prove $\sin^{-1}(1)\geq \int_0^b1/\sqrt{1-x^2}dx +(1-b)\pi/2$ for $b \in [0,1)$

I'm trying to prove the following inequality: $$\sin^{-1}(1)\geq\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$ for every $b \in [0,1)$. I'm given $\sin^{-1}(1) = \pi/2$ and $\sin^{-1}(x)$ is strictly ...
0
votes
1answer
47 views

How to find/construct functions $f(n)$ and $g(n)$ such that $\lim_{n\to \infty} f(n)e + g(n) = 0\:?$

Here, all $f(n)$ and $g(n)$ mentioned, are functions that assume integers values for some integer $n>0$. If $n>0$ and integer, we have that $I_n = \int_0^1 x^ne^xdx = f(n)e+g(n)$ where $f(n)=(-...
0
votes
0answers
14 views

Image of measure is relatively compact

Let $B$ be a Banach space and $\mu$ a $B$-valued measure on $X$. If $f: X \to B$ is a $\mu$-integrable function, then define $$\mu_f(E) = \int_E f d\mu$$ Is it true that the image of $\mu_f$ is ...
8
votes
0answers
66 views

Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$

Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$ Show that there exist $[a,b]\subset [0,1]$, such that $$\int_{a}^{b}f(x) dx= \int_{...
-1
votes
2answers
46 views

Prove that $u_n =\frac{n}{n^2+1}$ is a Cauchy sequence [on hold]

Please coulde you help me in this questions: Prove that $u_n =\frac{n}{n^2+1}$ is a Cauchy sequence and $v_n=\frac{n^2}{n+1}$ is not a Cauchy sequence.
0
votes
0answers
24 views

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable This is a practice problem for an exam I have coming soon. I am trying to study, but I ...
1
vote
2answers
41 views

Is this function continuous everywhere or at 1 point only?

Problem: let $f:[0,1]\rightarrow R$ be defined by $f(x) = \left\{ \begin{array}{lr} 2x-1 & : x \notin \mathbb{Q}\\ x^2 & : x \in \mathbb{Q} \end{array} \right.$ ...
1
vote
1answer
30 views

$\int_0^{\infty}\frac{\sin x}{(1+x)^2}\,dx$ converges absolutely

This is part of problem from baby Rudin Ch 6, exer 9: How to show that $$\int_0^{\infty}\frac{|\sin x|}{(1+x)^2}\,dx$$ exists (i.e., $\lim\limits_{a\to\infty}(\int_0^{a}\frac{|\sin x|}{(1+x)^2}\,dx)...
0
votes
1answer
18 views

Showing that a function has a minimum on a non compact interval

The question states: Let $f: (-1,1) \rightarrow R$ be cts and suppose $\lim_{x \to -1} f(x) = \lim_{x \to 1} f(x) = \infty$. Show that $f$ has a minimum on $(-1,1)$ My attempt: Given $\lim_{x \to -1}...
0
votes
2answers
39 views

sup, limsup, inf, liminf of $x_k=\frac{1}{k}+\cos(\frac{k\pi}{2})$ [check] [on hold]

I am considering the sequence $x_k=\frac{1}{k}+\cos(\frac{k\pi}{2})$ for $k\in\mathbb{N}$ for $\sup\{x_k\mid k\in\mathbb{N}\}$ I got $\frac{5}{4}$ (k=4) $\limsup_{n\rightarrow\infty}$ I got $1 $ $\...
-1
votes
1answer
67 views

Given that $\lim\limits_{n \to \infty} a_n \sum_{i=1}^n a_i^2=1$, Show $ \lim\limits_{n \to \infty} \sqrt[3]{3n}a_n=1$. [on hold]

Here is the problem: Given that $\lim\limits_{n \to \infty} a_n \sum_{i=1}^n a_i^2=1$, Show $ \lim\limits_{n \to \infty} \sqrt[3]{3n}a_n=1$.
0
votes
2answers
34 views

Differentiability of matrix-vector product map

Let $E$ denote the set of endomorphisms (linear operators) on $\mathbb{R}^n$, and define $G : \mathbb{R^n} \times E \rightarrow \mathbb{R}^n$ by $$ G(x,X) = Xx. $$ Question: Is $G$ differentiable? If ...
0
votes
1answer
24 views

Is this sum less when it contains less positive terms?

Problem Let $\{X_n\}$ be a sequence of real numbers defined by $$x_n= \left\{ \begin{array}{ccrcl} {\displaystyle\frac{1}{2k}} & \mbox{if} & {\displaystyle n} & {\...
0
votes
0answers
23 views

Explicit Injection of real-valued sequence space to real values

Injection requires that for some mapping $f : \mathbb{R}^\mathbb{N} \to \mathbb{R}$, $f(x) = f(y) \implies x = y$. I managed to show the existence of one by using a series of equivalences: $\mathbb{...
8
votes
4answers
126 views

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

So I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. So ...
4
votes
1answer
40 views

On Harmonic numbers at half-integer values

Harmonic numbers are usually defined, for $n\in\Bbb N$, by $$H_n=\sum_{k=1}^{n}\frac1k$$ But then one may note, $$H_n=\sum_{k=1}^{n}\int_0^1x^{k-1}\mathrm dx=\int_0^1\frac{1-x^n}{1-x}\mathrm dx=\int_0^...
2
votes
2answers
33 views

Why can I not use this alternative, simpler way of showing that $\frac{1}{|a|}f(\frac{x-b}{a})$ is Borel measurable

Let $X$ be a real random variable and $f$ is its density w.r.t. the Lebesgue measure. As a background, I was asked to show the density of $Y:=aX+b$ exists and is $g(x):=\frac{1}{|a|}f(\frac{x-b}{a})$....
-2
votes
1answer
20 views

As I can show by definition epsilon deltra that the exponential function is continuous at point 0 [on hold]

I think I should analyze when x> 0 and when x <0 but I'm not sure a delta comes to me: ln (e + 1) and the other ln ((1/1-e))
1
vote
3answers
59 views

Show that $E(Y^p)=\int_{0}^{\infty}px^{p-1}P(Y\geq x)dx$

Studying some probability theory and I came across this question. Show that if $Y$ is a non-negative random variable, and $p>0$, $$E(Y^p)=\int_{0}^{\infty}px^{p-1}P(Y\geq x)dx$$ I'm a bit stuck on ...
0
votes
1answer
26 views

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$. Hi, I need help with the above problem. I'm working ...
0
votes
4answers
78 views

Convergence of $\sum_{n=0}^{\infty}(-2)^{n^2}/n!$ [closed]

Is this series convergent $$\sum_{n=0}^\infty\frac{(-2)^{n^2}}{n!}\,?$$ I am supposed to find out if this series is convergent, absolutely convergent or divergent. No test has given me information. ...
-1
votes
0answers
22 views

Limit of a sequence within a sequence within a sequence

I'm working on a problem that involves a general sequence that I've been able to decompose two levels down (i.e. I have a sequence-within-a-sequence-within-a-sequence). I am aware that this general ...
0
votes
2answers
38 views

Can someone give me an example of a function $f$ being analytic but its power series $\sum_{n=0}^{\infty} a_{n}x^{n}$ diverging for some $x$?

is it necessarily true that $f$ being analytic implies its power series converges for all $x$? I think that it cannot diverge; however, I'm not very good at coming up with counterexamples. Can someone ...
0
votes
0answers
17 views

Frechet derivative of a function on matrix-valued $L^2$ functions

$\newcommand{\tr}{\text{Tr}}$ Let us denote $L^2(X, \mathbb C^{n \times n})$ be the space of matrix-valued $L^2$ functions. That is, if $f \in L^2(X, \mathbb C^{n \times n})$, each entry of $f$ is an $...
0
votes
3answers
23 views

Finding this Power Series's Interval of Convergence

We consider the power series $\sum_{n=1}^{\infty} a_nx^n$ where $a_n = \frac{2^k}{k}$ if $n$ is even, and $0$ otherwise. I understand how to find that this series has radius of convergence $R = \frac{...
-1
votes
1answer
24 views

Determine whether if the following sequences converge or diverge? [on hold]

$$ a_n= \frac{(-1)^n}{n^\frac32} $$ $$ b_n = 5 + (-1)^n(2 + \frac3n ) $$ show that $\sum a_nb_n $ converges. I know that $a_n$ converges by alternating series as $a_n$ but I don't understand how $...
1
vote
2answers
23 views

If $u(x)$ is harmonic and equal to $\phi(|x|)$, is $\phi$ continuously differentiable?

I was trying to show that radial harmonic functions on the unit ball (in $\mathbb{R}^n$) are constant. To this end, I suppose that $u$ is a radial harmonic function on the unit ball and write $$ u(x) =...
1
vote
2answers
45 views

Prove that the function $f_{n}(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$ converges pointwise for $x\in \mathbb{R}$.

I want to show that the function $$f(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}} $$ converges pointwise for all $x\in \mathbb{R}$. Furthermore, there are some intervals $(a, b)$ on which the function ...
3
votes
2answers
55 views

uniformly continuous function $f$ such that $\sum 1/f(n)$ is convergent?

Does there exist a uniformly continuous function $f:[1,\infty)\to \mathbb R$ such that $\sum_{n=1}^\infty 1/f(n)$ is convergent ? I know that $\exists M>0$ such that $|f(x)|< Mx, \forall x\in ...
2
votes
0answers
20 views

Taking a limit involving a function of bounded variation.

Let $f:[0,1] \rightarrow \mathbb{R}$ be a function of bounded variation. Let the total variation of $f$ be $a$. Let $E = \{ x \in (0,1): \text{lim sup}_{h \rightarrow 0} \frac{|f(x+h)- f(x)|}{|h|} &...
1
vote
0answers
103 views

Isometry between two $L^p$ spaces

I have a measurable space $(X,M,\mu)$ and $\mu$ is $\sigma-$finite, then there exist a finite measure $\lambda$ s.t. the space $L^p(\mu)$ is isometric to $L^p(\lambda)$. First I proved that there ...
3
votes
2answers
38 views

Showing that $M := \{f \in C([-1,1]) : f(0) > 0\}$ is open

Given $f \in C([-1,1],\mathbb{R})$ equipped with sup norm metric. I am trying to find out whether the subset $$M := \{f \in C([-1,1]) : f(0) > 0\} $$ is open/closed in $C([-1,1])$ My work: $M$ is ...
0
votes
4answers
110 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.
5
votes
0answers
94 views

Decay of positive definite functions in Lp

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means $$ \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0 $$ for all $...
0
votes
1answer
59 views

Is this true that the sequences evenly sampled from functions in $L^p$ are in $l_p$?

Is the following statement true ? Given a function $f(x)$ in $L^p(\mathbb{R})$, $1\le p<\infty$, the sequence $\{f(x+kT)\}_{k\in\mathbb{Z}}$ is in $l_p$ for any $x$ and $T$ if every number in ...
0
votes
0answers
41 views

If $f'$ is continuous at the point $x_0$, does this imply $f$ is Lipschitz around $x_0$

Let $f : R^n \to R^m $ which is differentiable around the point $x_0$, whose derivative is continuous at the point $x_0$. Does this imply $f$ is Lipschitz around $x_0$ My intuition: Since $f'$ is ...
1
vote
3answers
77 views

Taylor series of $\ln\frac{1+x}{1-x}$ [duplicate]

Let $f(x)=\ln\frac{1+x}{1-x}$ for $x$ in $(-1,1)$. Calculate the Taylor series of $f$ at $x_0=0$ I determined some derivatives: $f'(x)=\frac{2}{1-x^2}$; $f''(x)=\frac{4x}{(1-x^2)^2}$; $f^{(3)}(x)...
8
votes
1answer
117 views

Is there a continuous waveform that sounds the same as a square wave?

The fourier series $$f(t)=\sum_{n\in\mathbb N\\n\text{ odd}}\frac1n\,\sin(nt)$$ converges to a square wave. Square waves are discontinuous functions. I'm wondering if there's a continuous function ...