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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
3answers
24 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
vote
2answers
42 views

Is this function continuous everywhere or at 1 point only?

Problem: let $f:[0,1]\rightarrow R$ be defined by $f(x) = \left\{ \begin{array}{lr} 2x-1 & : x \notin \mathbb{Q}\\ x^2 & : x \in \mathbb{Q} \end{array} \right.$ ...
0
votes
2answers
41 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 $ $\...
3
votes
2answers
64 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 ...
1
vote
2answers
82 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+\...
2
votes
0answers
22 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
2answers
30 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) =...
2
votes
2answers
39 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 $g(x):=\frac{1}{|a|}f(\frac{x-b}{a})$....
3
votes
2answers
39 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
0answers
45 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
votes
0answers
24 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
0answers
28 views

Calculus-probability

I have been working on a problem that needs to be proved. Then, in the middle of the proof I was blocked because I could not upper bound this probability given bellow. Actually, the double sum of the ...
1
vote
0answers
18 views

How to show convex hull of finitely many vectors in $\mathbb{R^n}$ is a compact set? [duplicate]

Suppose $x_1,x_2,\cdots,x_k$ is any finite set of vectors in $\mathbb{R^n}$. A convex combination of these vectors is $\sum_{i=1}^{k}\lambda_ix_i$ where $\sum_{i=1}^{k}\lambda_i=1$ and $\lambda_1,\...
1
vote
2answers
49 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 ...
0
votes
0answers
19 views

Continuity on infimum defined function [on hold]

Let $f(x)$ be continuous in $[a,b]$. Consider the function $m(x)= \inf\{f(t):$ $t$ belongs in $[a,x]\}$. Prove that $m(x)$ is continuous
1
vote
1answer
35 views

How to show the function $f(x) = x^2 \sin(1/x)$ has integrable derivative?

Consider the function $$f(x) = \begin{cases} x^2\sin(1/x) & \text { if } x \neq 0 \\ 0 & \text{ otherwise} \end{cases} $$ has integrable derivative on $(-1, 1)$. I found $$f'(x) = \begin{...
4
votes
1answer
43 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^...
1
vote
0answers
16 views

If the Taylor expansion of $f$ converges to $f$, prove that there are constants $C,R$ such that $f^{(k)}(x) \le C \cdot \frac{k!}{R^k}$

Q: Let $f$ be an infinitely differentiable function on an open interval $I$ centered at $a$. Assume that the Taylor expansion of $f$ about $a$ converges to $f$ at every point of $I$. Prove that there ...
3
votes
5answers
68 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
20 views

Monotonicity of tangent

obviously for $a,b \in ]\frac{-\pi}{2},\frac{\pi}{2}[$ the ordinary tangent map is strictly monotonic. Hence for $b \geq a \Rightarrow \tan(b) \geq \tan(a)$. In the proof I try to understand it ...
1
vote
3answers
80 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)...
0
votes
1answer
25 views

Proving convergence of a sequence with logarithm

I was wondering about the convergence of a sequence based on the convergence of the logarithm of the sequence. I can't seem to find anything online, and it seems to make intuitive sense, but I'm not ...
0
votes
2answers
54 views

An application of mean value theorem

Let $f:[0,1] \to \Bbb{R}$ a differentiable function and $M>0$ such that $f'(x) \geq M,\forall x \in [0,1].$ Prove that exists an interval $I$ with length $\frac{1}{4}$ such that $|f(x)| \geq \...
0
votes
0answers
36 views

Hint Prove $||b|^p-|a|^p-|a-b|^p|\leq C_{p}(|a|^{p-1}|a-b|+|a||a-b|^{p-1})$

I have looked at this inequality in several ways, but cannot find the correct path: Show for $a,b \in \mathbb R$ and $C_{p} > 0$ with $p \in ]1,\infty[$ $||b|^p-|a|^p-|a-b|^p|\leq C_{p}(|a|^{p-1}|...
8
votes
1answer
130 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 ...
1
vote
1answer
33 views

Sequence with anchoring property is convergent in finite dimension

Let $(x_k)$ be a sequence in $\mathbb{R}^n$. Furthermore, assume that the sequence has the following "anchoring" property: $\exists y,z \in\mathbb{R}^n$ such that $\lim|x_k-y|=r_y$ and $\lim|x_k-z|=...
0
votes
3answers
38 views

Show that two primitives/ antiderivatives are related via a constant

Let $I \subset R$ be an interval. A differentiable function $F: I \rightarrow R$ is called a primitive for the function $f : I \rightarrow \mathbb{R}$ if $$F' (x) = f(x)$$ for all $x \in I$. ...
2
votes
1answer
60 views

Reimann zeta function

I started solving a problem using the transformation of Reimann zeta function into this form: And I searched for the methods of doing this transformation and ended up with this one : It is a little ...
14
votes
1answer
259 views
+50

How to solve $\dot{x}=f(x)/||f(x)||$?

How to solve the following ODE? \begin{equation} \frac{d}{dt}x=\frac{f(x)}{\|f(x)|\|}, \end{equation} where $x: \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f:...
-3
votes
2answers
33 views

Sum of reciprocals of powers of two with alternating signs. [on hold]

What is the value of $1 - \frac{1}{2} - \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} - \frac{1}{8} - \frac{1}{8} - \frac{1}{8} - \frac{1}{8} - \frac{1}{8} - \frac{1}{8} - \frac{...
3
votes
0answers
34 views

Are functions satisfying $|f(x)-f(y)|\le L \|\nabla f(x)-\nabla f(y)\|^{1+s}$ constant?

Let $f:\mathbb R^n\to \mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such $$ |f(x)-f(y)|\le L \|\nabla f(x)-\nabla f(y)\|^{1+s} \quad \forall x,y\in\mathbb R^n. $$ ...
0
votes
0answers
18 views

Interchanging limit and expectation in Ito Isometry proof

Let $\phi_n$ be a sequence of elementary functions and $f$ a function satisfying: $f(t,\omega)$ is measurable wrt $\mathcal{B}\times \mathcal{F}$ $f(t,\omega)$ is $\mathcal{F}_t$-adapted $E\left[\...
6
votes
2answers
128 views

What exactly ARE $\pi$ and $e$? [on hold]

First of all, apologies if this is a bad question. I don't really know how to phrase it. I first got introduced to $\pi$ in elementary school, where it was presented as a ratio for a circle's area. ...
-1
votes
1answer
33 views

If f is not Riemann integrable on an interval can the absolute of f be Riemann Integrable on that interval? [on hold]

If f(x) is not Riemann integrable on some interval say I, can the absolute value of f(x) be Riemann integrable on that interval I ?
3
votes
2answers
88 views

Prove $\sum\limits_{n=1}^{\infty} a_{f(n)} = \dfrac{3}{2} \ln(2) $

Where $a_n = \dfrac{(-1)^{n+1}} { n}$ and $f:\mathbb{N}\to \mathbb{N} $ is a bijection which for $k \in \mathbb{N}\setminus \{0\}$ is given by: $$f(3k+1)= 4k+1$$ $$f(3k+2)= 4k+3$$ $$f(3k+3)= 2k+2.$$...
1
vote
1answer
72 views

Proof from definition that $\lim_{x\to \infty} e^{-x^2}e^{2x}=0$

I'm trying to give a proof from definition that $\lim_{x\to \infty} e^{-x^2}e^{2x}=0$ I know $|e^{-x^2}e^{2x}|<|e^{2x}|$. From there trying to get $|e^{2x}|<\varepsilon$, but that isn't useful ...
2
votes
1answer
35 views

$\lim_{n \to \infty} (-1)^n !ne+ (-1)^{(n+1)} n! = 0 \: ?$

$\lim_{n \to \infty} (-1)^n !ne+ (-1)^{(n+1)} n! = 0 \: ?$ $!n$ is the subfactorial. Here's a plot made with wolfram: Maybe the function goes to $0$ at infinity. How to prove this analytically? I ...
1
vote
2answers
32 views

Borel measure determined uniquely on a base

Let $(X,\tau)$ be a topological space, $\mathcal{B}$ the $\sigma$-algebra generated by $\tau$ on $X$. Let $\mu$ be a Borel measure on $X$. Does the restriction of $\mu$ to a base for the topology $\...
1
vote
1answer
45 views

Question about strong convexity

I don't really know how to begin. I tried substituting $y$ for $x + h$ and taking the Taylor approx. of $f(x + h)$ around $f(x)$. The RHS becomes $h^T \nabla f(x) + \phi(x)$ Where $\phi$ is our ...
0
votes
1answer
26 views

Infinite linear combination

I don't know how to start the proof of the following statement: Let $V$ a real vector space with inner product $\langle\cdot,\cdot\rangle$. Consider a subset $C=\{v_j\}_J\subset V$ such that $\...
0
votes
1answer
22 views

Ito isometry and convergence in $L^2 (P)$?

I am just wondering if my understanding of the importance of Ito isometry is correct. The Ito isometry says that for $f$ satisfying certain conditions, we have $$E\left[\left(\int_S^T f(s,\omega) ...
-4
votes
0answers
42 views

Prove that E=[0,1] is closed on $\mathbb{R}$. [on hold]

I know that I need to show that $E^c$=$(-\infty,0) \cup (1,+\infty)$ is open but am having a hard time figuring out how to do that. Could someone please walk me through the steps?
-1
votes
0answers
46 views

Is average operator continuous? [on hold]

Suppose $$Lf=\frac{1}{x}\int_0^xf(t)d\mu(t)$$ where $d\mu(t)=\frac{1}{t}dm(t), dm$ is the Lebesgue measure. Is it true that $L$ is continuous in $L^p((0,\infty),\mu)$, $1\leq p\leq\infty$ ?
0
votes
1answer
43 views

What does it mean for a vector function to be twice differentiable at a point?

Let $F: R^n \to R^m $ be a vector valued function with components $F = (f_1 ,f_2 ,..., f_m).$ What is the most proper and universal accepted definition for the function $F$ be twice differentiable ...
0
votes
1answer
30 views

Tchebyshev inequality for Measure theory.

Suppose $f$ integrable on measurable set $E$ and $f(x)$ $\geq$ $0$ on $E$, And for any $a \in \mathbb{R}^+$, we define: $E_a$ = $\{x \in E$ $\vert$ $f(x) > a\}$. (which is measurable by basic ...
4
votes
0answers
66 views

Counterexample to Riemann sum limit

For convergent Riemann sums of $f \in C^1([0,1])$ there is the property: $$\tag{A}\lim_{n \to + \infty} \left[\, \sum_{k=0}^{n} f \left( \frac{k}{n+1} \right) - \sum_{k=0}^{n-1} f \left( \frac{k}{n} \...
1
vote
1answer
31 views

Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Let $\{a_n\}_{n \geq 1}$ be a non-negative sequence of reals such that $\sum_{n \geq 1} a_n$ converges to $s$. Define $N_k = |\{n \in \mathbb{N} : a_n \geq 2^{-k}\}|$. Show that \begin{equation} \...
0
votes
1answer
35 views

Prove inverse of strictly monotone increasing function is continuous over the range of original function

Let $f:[a,b] \rightarrow \Bbb R$ be a strictly monotone increasing. Then $f$ has an inverse function $g:[c,d]\rightarrow \Bbb R,$ where $[c,d]$ is the range of $f$. I'm trying to prove that $g$ is ...
0
votes
0answers
43 views

What's a good exercise for introductory real analysis? [on hold]

I have a final tomorrow, and I need some more exercises. Anything similar to Baby Rudin chapter 4 and 5. Thanks
1
vote
1answer
13 views

Given interpolating function on an interval, find the upper bound of the error for that function

Let p(x) be a linear function interpolating sin(x) at $x=0, x=\frac{\pi}{2}$. Prove that $|p(x)-\sin x| \leq \frac{1}{2}(\frac{\pi}{4})^2$ on $[0, \frac{\pi}{2}]$. I've already done a bit of work and ...