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).

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1answer
12 views

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$.

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$. What I have: Assume that $k<0$, need to show that this contradicts the ...
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1answer
25 views

What can we say about the definite integral and Riemann sums?

Consider $$f(x) = \int_{1}^{x} \frac{dt}{t}$$ and $$g(x) = \sum_{1}^{x} \frac{1}{t}.$$ I would like to say that $f(x) < g(x)$ for all natural numbers $x \ge 1$. Is there an easy way to derive ...
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0answers
21 views

Determine the limit of a series, involving trigonometric functions.

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $$\sum^\infty_{n=1} \frac{\sin((n+1)x)}{(n+1)^3}-\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}≤\sum^\infty_{n=1} ...
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0answers
5 views

Riemann Sum of a sequence of partitions proof

Given that $\lim_{||P|| \rightarrow 0} S(P, f) = I$ (where I is in $\Bbb R$ and S(P,f) is the Riemann sum of f). How do I show that if $(P_n)_{n \epsilon N}$ is a sequence of partitions with $||P_n|| ...
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0answers
16 views

Weak star convergence question

Let $C$ be a convex cone in $L^{\infty}$, that is if $x,y \in C$ and $\alpha, \beta > 0 $ then $\alpha x + \beta y \in C$. Let $U$ be the unit ball in $L^{\infty}$. Assume that for each sequence ...
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2answers
63 views

$ \int \frac{dx}{4x^2-12x+13}$

This is probably not too hard but i can't get it right: I am trying to calculate $$\displaystyle \int \frac{dx}{4x^2-12x+13}$$. The solution is $\displaystyle ...
1
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1answer
34 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
1
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1answer
32 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
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0answers
9 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
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1answer
38 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
1
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1answer
12 views

Proof via strong induction of a string output

I'm still new to the whole proof thing (first class of discrete mathematics and analysis right now). I could do general induction problems, but the fact that 'n' is the output here along with the ...
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0answers
15 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
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2answers
41 views

Jacod Protter “Probability Essentials” Problem 2.8

The question asks to show that a sigma-algebra $\mathcal A$ consisting of $A$ s.t. $A=f^{-1}(B)$, where $B$ is in $\mathcal B$ are Borel subsets of $R$ and $f$ is continuous, is contained in $\mathcal ...
1
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1answer
26 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...
0
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1answer
21 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
0
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1answer
17 views

$\lim_{n \to \infty} \int_{\Omega}X_n d \mu = +\infty$ under some conditions

Suppose $X_n$ are measurable functions in $L^1$ defined on the measure space $(\Omega, \mathfrak{F}, \mu)$. Suppose that $0 \leq X_n$ a.e. for all $n$ and $X_n \leq X_{n+1}$ a.e. for all $n$. Thus ...
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0answers
21 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
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0answers
13 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
3
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1answer
51 views

False equations with Euler's Identity [duplicate]

What's wrong with the following equations? $$1 = 1^{-i} = (e^{2πi})^{-i} = e^{-i2πi} = e^{2π}$$ My guess would be the third equation, but I can't really tell why... in the first equation, we use the ...
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0answers
47 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
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1answer
31 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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1answer
18 views

Application of the IVT

Is it true that on any circle there is a pair of opposite points where the age of the surface rock is the same? I think the answer is no. In the temperature case the function T: [ 0, 2π] → R where ...
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4answers
54 views

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$. What I have: Since $\{a_n\}\rightarrow \alpha$ we know that ...
0
votes
1answer
28 views

Bounding summations

Show that $\sum k2^k = \Theta( k2^k)$. I tried to use mathematical induction to prove the bound, but it didn't work. There are other ways that can be used to prove this bound, like bounding the ...
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3answers
73 views

Proof of an inequality by induction

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
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0answers
21 views

Lebesgue integral of cartesian product of functions

Given two Lebesgue Integrable functions $f,g$, is there a notion of the integral $$\int_A f \times g \, \, dx_1 \times dx_2 ?$$ Is this even a definable notion? I couldn't find anything on the ...
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0answers
27 views

Multiplication on Reals as equivalence classes of cauchy sequences is well defined

So I understand the solution for the proof that multiplication of equivalence classes of cauchy sequences is well defined using boundedness of cauchy sequences and a chain of inequalities. I just ...
0
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2answers
27 views

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$. What I have: Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ ...
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0answers
36 views

Inflection point and 2nd derivative

Is it possible a function $f:\mathbb{R} \rightarrow \mathbb{R}$ to have an inflection point somewhere but that it is not two times differentiable at that point? If so, then can we have a form of that ...
2
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2answers
44 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
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0answers
13 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
4
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1answer
29 views

If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.

This is what i know; If $(a_n)$ is an infinite sequence of which is bounded then we can say; $|a_i| < M $ for all $i \geq 0.$ since $a_n$ and $b_n$ are equivalent sequences, we can say that for ...
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2answers
77 views

How do I solve this Olympiad question with floor functions?

Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part ...
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0answers
54 views

Terrence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.

Let $ x = \lim_{n\rightarrow\infty}a_n, y = \lim_{n\rightarrow\infty}b_n$, and $ x' = \lim_{n\rightarrow\infty}a'_n$ be real numbers. Then $xy$ is also a real number. Furthermore, is $x=x'$, then $xy ...
0
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2answers
56 views

Prove that $(f_n)_n$ is uniformly convergent.

Let $g$: $[0,1]\to\mathbb{R}$ be continuous and $g({1})=0$. Define $f_n(x)= x^{n}{g(x)}$. Prove that $(f_n)_n$ is uniformly convergent.
1
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1answer
30 views

prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$

The real function ${g}$ is continuous on $[0,1]$ .we define ${f_n}$ on ${[0,1]}$: $$f_n(x)=\frac{{{g(x)\sin^{n} (x)}}}{{{1+nx}}}$$ prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$ .
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1answer
25 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
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0answers
17 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
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1answer
42 views

absolutely convergent & conditionally convergent [on hold]

Prove that $$\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n}\frac{{{1}}}{{{m}}}\right)\frac{{{\sin(nx)}}}{{{n}}} $$ for $x = {k\pi}$ , $k\in \mathbb{Z} $ is absolutely convergent. & for $x ...
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0answers
78 views

Explain about absolute convergence and convergence [on hold]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
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2answers
57 views

Prove$\sum_{n=0}^{\infty} \frac{n}{(n+1)^2}-\frac{1}{n+2} $ is convergent [on hold]

prove $$\sum_{n=0}^{\infty}\frac{{{n}}}{{{(n+1)^{2}}}}-\frac{{{1}}}{{{(n+2)}}} $$ is convergent.
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0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
0
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2answers
43 views

$ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent. [on hold]

prove that if $ \sum_{n=1}^{\infty} {a_n} $ is absolutely convergent $ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent.
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1answer
22 views

prove that Radius of convergence is 1 [on hold]

Let's assume that $${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$$ is a positive sequence number. and let $$ \mathop {\lim }\limits_{k \to \infty }{A_k}=\sum_{n=0}^{k} {a_n} = \infty $$ if $$ \mathop ...
0
votes
1answer
26 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
1
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1answer
13 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
2
votes
0answers
14 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
1
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0answers
22 views

Local existence for semilinear wave equations

After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} ...
-2
votes
1answer
39 views

How can I evaluate the definite integrals with limits?

How can I evaluate the following limit $$\lim_{n\to\infty}\int_{0}^{1}\left(\frac{x^2+x+1}{4}\right)^n\sin(nx)dx$$
1
vote
1answer
21 views

How do I compute this Milnor number

I need to compute $\mu (x^5+y^5)=5$ on the point $p=(0,0)\in\mathbb{C}^2$. By definition, for $f\in\mathbb{C}[x,y]$, I have $$ \mu(f)=\dim\dfrac{\mathcal{O}_{(0,0)}}{<\dfrac{\partial f}{\partial ...