Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

learn more… | top users | synonyms (1)

3
votes
1answer
74 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...
0
votes
1answer
61 views

Approximating a characteristic function and derivative

Let $f(x) = \chi_{[0, 1)}$. Is there a sequence of smooth functions $f_{n}$ such that $\int_{0}^{1}|f_{n}'(t)|\, dt \rightarrow 0$ and $\|f_{n} - f\|_{L^{1}} \rightarrow 0$ as $n \rightarrow \infty$.
4
votes
3answers
144 views

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ? The numerator is a irreducible polynomial so I can't use partial fractions. I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int ...
1
vote
0answers
77 views

Relationship between n-dimensional ellipsoid's surface area and semi-axes

I'm trying to prove (or falsify) the following claim relevant to work I'm doing with Steimer Symmetrization: If we consider an n-dimensional ellipsoid with semi-axes of radius $a_1< ... < a_n$, ...
0
votes
2answers
48 views

Small question about limit

if i have $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)-a|u|^{\tau-2}u}{u}=0$ how to prove that $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)}{|u|^{\tau-2}u}=a$ such that $\tau\in (1,2)$ I ...
0
votes
1answer
32 views

Manifold and scalar product defined as an integral

Let define $B=\{x_1^2+...+x_n^2 \leq 0\}$ with boundary $S=\{|{x^2}|=1\}$. Then let $X=\{f\in C^2(B):f|_{S}\equiv0\}$ be a space with scalar product given as follows: $$(f,g)=\int_B fg\ d\lambda_n\ ...
2
votes
3answers
95 views

How to evaluate $\lim\limits_{n\to\infty}(n^3)\cdot(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt{2})$?

How to evaluate $\lim\limits_{n\to\infty}(n^3\cdot(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt{2}))$? My though is, that since $\lim\limits_{n\to\infty}n^3=\infty$, the limit must be $\infty$, if the right ...
0
votes
1answer
39 views

General approaches to test convergence of $\sum a_n z^{n^p}$ when z is a complex number of unit length 1

Suppose $|z| = 1 $ for some complex $z$, I know $\displaystyle \sum z^{n}$ diverges since by summing the geometric sequence, but what can we say about things like: $\displaystyle \sum z^{n^p}$ ...
2
votes
1answer
65 views

Extending solutions of an ODE past a singular point

In the course of my studies, I'm looking at at the ODE: \begin{equation} (f^3(x))'''=\frac{1}{6}xf(x),\quad f(0)=1,\,\,f'(0)=0 \end{equation} Where $f''(0)$ is a parameter left undetermined. In ...
1
vote
1answer
39 views

Sturm Liouville problem with additional term.

Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$ $f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue. Furthermore, ...
0
votes
0answers
36 views

Show that Intersection and union of Ka is compact

Let $ \{ K_a \} $ be a collection of compact subsets of a metric space $(X,d)$. a) Show that $ \cap_{a=1}^\infty K_a$ is compact. b) Give an example showing that $\cup_{a=1}^\infty K_a$ is not ...
2
votes
1answer
43 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of ...
28
votes
2answers
557 views

Can we find this infinite root in term of elementary function? [closed]

Let $f(x)=\left(x+f(x+1)\right)^\frac{1}{x}$. What is the value of $f(2)$ ? More precisely, how to find the value of $$\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}~?$$ Thank you.
0
votes
2answers
21 views

pointwise convergence, how can I show that $f_n \not\to 0$ uniformly?

I am given the following exercise: Let $\displaystyle{f_n:(0,+\infty) \to \mathbb{R}, f_n(x)=\frac{\sin(nx)}{nx}}$. Show that $f_n \to 0 \text{ pointwise }, x \in (0,+\infty) \text{ and } that f_n ...
3
votes
4answers
1k views

$f_n(x):=nx(1-x)^n$ Determine whether the sequence (f_n) converges uniformly on $[0,1]$

I am having a bit of trouble on this revision question. To determine pointwise convergence: $\lim_{n\rightarrow\infty} = nx(1-x)^n $. For $x=0, x=1$, it's clear that the limit is $0$. How can I ...
1
vote
3answers
46 views

Are these roots the only ones or are there more roots?

I am looking at the following exercise: Let $f:[a,b] \to \mathbb{R}$ integrable. Show that $\exists \xi \in [a,b]$ such that $\int_a^{\xi} f(u) du= \int_{\xi}^{b} f(u) du$ We define ...
2
votes
3answers
68 views

$f_n(x):= \frac{nx^3}{1+nx^2}$. Show that the sequence $(f_n)$ converges uniformly on $\mathbb{R}$

$f_n:\mathbb{R} \rightarrow \mathbb{R}$ A bit stuck on this revision question. I first determine the pointwise limit easy enough: $$\lim_{n\rightarrow \infty} \frac{nx^3}{1+nx^2} = x$$ To show that ...
5
votes
2answers
139 views

Divergence of the sequence $\sin(n!)$

Does the sequence $\sin(n!)$ diverge(converge)? It seems the sequence diverges.I tried for a contradiction but with no success. Thanks for your cooperation.
0
votes
1answer
114 views

What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
2
votes
1answer
135 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
1
vote
0answers
46 views

Classic applications of Baire category theorem [duplicate]

Problem: Suppose that $f:\Re^+\to\Re^+$ is a continuous function with the following property: for all $x\in\Re^+$, the sequence $f(x), f(2x),f(3x)\cdots$ tends to $0$. Prove that ...
1
vote
1answer
212 views

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
0
votes
1answer
123 views

Smooth function and mollifiers with $\mathcal{C}^{\infty}$ continuation to a given function

We consider a function $f : [0 ; 1] \mapsto \mathbb{R}$, which is $\mathcal{C}^{\infty}$ on the open interval, and positive. For example $f(x) = 1 + x$. ($f$ is a nice function, not a nasty beast.) I ...
0
votes
1answer
53 views

Exercise on isometry

Let $X$ be a Banach space and $T$ a linear bounded operator defined on $L(X,Y)$ with $Y$ a normed space. If $T$ is an isometry then $TX$ is a closed subspace of $Y$. I considered a sequence $y_n$ ...
1
vote
5answers
204 views

Complex integral - exercise

$$ \int\limits_{C(-2, \frac{1}{4})} = \frac{e^z}{z^2-4}dz$$ C is a circle center = -2 and radius = $\frac{1}{4}$ z is a complex number I don't know how to do the exercises like that.
0
votes
1answer
40 views

Prove that a sequence is decreasing

Suppose that 0 < a < 1 Show that {a^n} is a decreasing sequence. Yes, this is a homework question. I think I can solve it using induction, but I'm not sure.
1
vote
1answer
53 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
0
votes
1answer
52 views

Complex Polynomial That is n Times Differentiable: A Concern

I'm looking at a question that asks me to show that: If a function $f$ is known to be $n$-times differentiable in a domain $D$ and if $\forall{z\in{D}}\ \ f^{(n)}(z)=0$, then $f$ is a polynomial ...
4
votes
1answer
166 views

What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $?

For every $ x,y \gt 0$, if $ xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
0
votes
1answer
54 views

Inner product on Hilbert Spaces

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?
6
votes
1answer
139 views

How to calculate the integration $\int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2}$ [duplicate]

Given that $$ \int_{0}^{\pi}\frac{dx}{2-\cos{x}}=\frac{\pi}{\sqrt{a^2-1}} $$ How to calculate the integral $$ \int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2} $$
0
votes
1answer
61 views

Dense Cantor set approximation

I am reading Measure, Topology and Fractal Geometry by Edgar and in the first few pages he defines the Cantor set and a dense approximation to the Cantor set. He says that $\frac{1}{4} = ...
1
vote
1answer
167 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
1
vote
1answer
51 views

Small question about strong convergence

I have a small question I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poincaré inequality i have that ...
1
vote
1answer
147 views

A problem involving Stieltjes Integral and bounded variation

I found this problem in a book I'm using to study (Curso de Análise - Vol 2, Elon Lages Lima). "Let $\alpha:[a,b] \to \mathbb{R}$ be a bounded function. If $\displaystyle \int_{a}^{b}f(t)d\alpha$ ...
2
votes
1answer
63 views

How to conclude that the Taylor series of $f(x)=\log(1+x)$ is equal to $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} (n-1)!}{(1+x)^n}$

I want to find the Taylor series of $f(x)=\log(1+x), x \in (-1,+\infty), \xi=0, I=(-1,1)$ $$f'(x)=\frac{1}{1+x}$$ $$\sum_{n=0}^{\infty} (-1)^n x^n=\frac{1}{1+x}, x\in (-1,1)$$ $$f(x)=f(0)+\int_0^x ...
2
votes
2answers
57 views

How to find the Taylor series of $f(x)=\arctan x$.

I want to find the Taylor series of $f(x)=\arctan x,\; x\in[-1,1],\;\xi=0$. That's what I have tried do far: $$f'(x)=\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=\sum_{n=0}^{\infty} (-x^2)^n.$$ How can I ...
3
votes
1answer
41 views

$\lim_{x \to 0} \dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} = 1$ for any $f,g \in C^1$ that are tangent to $\text{id}$ at $0$ with some simple condition

Theorem For any real functions $f,g \in C^1$ such that $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$ and $x$ is strictly between $f(x)$ and $g(x)$ for any $x \ne 0$:   $f,g$ are invertible on some ...
2
votes
3answers
59 views

Supremum of two subtracted fractions less than one

Let $$S=\left\{\frac{1}{n}-\frac{1}{m}: m,n∈\mathbb{N}\right\}$$ Find the Supremum of this set. I get the feeling that the answer is $1$ as if you let $n=1$ and $m$ be infinitely large then its ...
6
votes
3answers
182 views

If limit of $f(x)$ exists and the limit of $f(x)g(x)$ exists, then does the limit of $g(x)$ exist?

I was doing some exam preparation and I got stuck on this one. Whats the idea behind this question: If $\lim_{x\to a}f(x)$ and $\lim_{x\to a} [f(x)g(x)]$ exist then does $\lim_{x\to a}g(x)$ exist? ...
1
vote
1answer
90 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
3
votes
1answer
112 views

can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
0
votes
1answer
42 views

How do we find a path $\gamma$ with winding number $1$ and $2$ relative to points $1$ and $2$, respectively?

Let $\gamma :[a, b]\to\Omega\subseteq\mathbb{C}$ denote a parametric piecewise continuously differentiable path in $\Omega$ and $$\text{ind}_{\gamma}(z):=\frac{1}{2\pi ...
1
vote
2answers
48 views

Show that $f\in \mathcal{R}[0,1]$

Let $f$ be defined by $$f(x) =\left\{\frac{1}{n}, \frac{1}{n+1} \lt x \le \frac{1}{n} \right\}$$ and $f(x) = 0$ when $x=0$ and $n \in N$. Show that $f$ is integrable and ...
2
votes
1answer
75 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
0
votes
1answer
122 views

How to calculate a Frobenius norm?

Suppose that $A$ is an $n \times m$ ($n$ less than $m$) full rank matrix. Apply Gram-Schmidt orthogonalization to the rows of $A$, then we get an $n \times m$ matrix $B$ with orthonormal columns. ...
0
votes
2answers
34 views

Are there functions that converge to $P$ when $f:\mathbb{N}\to\mathbb{R}$ and to $Q$ when $f:\mathbb{R}\to\mathbb{R}$ with $Q\neq P$?

I remember that my professor gave one example of a function that when $f:\mathbb{N}\to\mathbb{R}$, this function converged to a number, say, $P$. And when $f:\mathbb{R}\to\mathbb{R}$, it also ...
0
votes
1answer
43 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
4
votes
3answers
109 views

Suppose that $f \in C[0,1]$ . Show that $\lim_{n\to \infty}\int_{0}^{1}(n+1)x^{n}f(x)dx=f(1)$

Suppose that $f \in C[0,1]$ . Show that $\lim_{n\to \infty}\int_{0}^{1}(n+1)x^{n}f(x)dx=f(1)$. This is how I proceeded: Suppose $x^{n+1}=t$ . Then $(n+1)x^ndx=dt$. Now the integral becomes ...
4
votes
2answers
146 views

How to calculate factorial function as $x\to\infty$?

I need to calculate $$\lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}.$$ Even I used Striling Approximation and Wolfram Alpha, they do not help. How can I calculate this? My ...