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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96 views

Deduce that $\sum \frac {a_n}{r_n}$ diverges.

Suppose $a_n>0$ and $\sum a_n$ converges. Put $r_n=\sum_{n=m}^\infty a_m$. Prove that $$\frac {a_m}{r_m}+\dots+\frac {a_n}{r_n}>1- \frac {r_n}{r_m},$$ if $m<n$, and deduce that $\sum \frac ...
2
votes
3answers
121 views

Different limits for the alternating harmonic series?

Show that the series $$\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n}$$ is not absolutely convergent. Show that by permuting the terms of the series one can obtain series with different limits. I am able to ...
4
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2answers
165 views

Prove that, if $g(x)$ is concave, for $S = {x : g(x) > 0}$, $f(x) = 1/g(x)$ is convex over $S$.

Using the definitions of convexity and concavity, I need to show the following: $$g(ax + (1-a)y)\geq ag(x) + (1-a)g(y),\ \ a \in (0, 1)$$ implies that $$f(ax + (1-a)y) \leq af(x) + (1-a)f(y)\ , a ...
0
votes
1answer
127 views

Difficulties in grasping the proof of the Heine-Borel theorem

I’m finding it difficult to grasp a few nuances of the proof of the Heine-Borel theorem as laid out in my textbook. Here is the passage that is giving me problems: “Suppose a set S is closed and ...
1
vote
2answers
201 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
1
vote
1answer
34 views

Let $y_1,…,y_k$ be the roots of $q$. Why is $q(x)\prod_{i=1}^n(x-y_i)$ only positive or only negative.

I'm trying to understand this exercise: Well, my teacher told me that I need to suppose $q$ has $k<n$ different roots in $(a,b)$. So we have the roots $y_1,...,y_k$ of $q$. Then if I set $p(x)= ...
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0answers
38 views

Check for the value of x for function f is ill conditioned

I have two examples: $$ i) \ f(x) = \sqrt{1 - x^2} $$ $$ ii) \ f(x) = \sqrt{x^2 + 1} - x $$ I must check the value of x for which the calculation of the values ​​of the function $ f $ is ill ...
1
vote
1answer
173 views

Show that a set is measurable with respect to Borel product $\sigma$-algebra

I'm having some trouble with the following exercise: Let $(\mathbb{R}, \mathcal{B})$ denote the real line with the Borel $\sigma$-algebra and let $X=(\mathbb{R}, \mathcal{B})\times(\mathbb{R}, ...
2
votes
1answer
98 views

Showing continuity

Let $|f(x)|\le \dfrac{A}{1+x^2}$ for all $x$ and some $A$ (to ensure that $\int_{-\infty}^\infty f(x)\rm{d}x$ makes sense). I would like to show that $$g(z) = \int_{-\infty}^t f(x) e^{-2 \pi i z ...
2
votes
3answers
107 views

Are there any whacky orderings of R?

Is there any way to reorder R so that 3 < 2? And a similar question, which probably can be answered in the same breath: Is d(2, 3) < d(2, 100) for all metrics? Is there a nice theorem that ...
1
vote
1answer
81 views

About the continuity of a convolution product

I need some help with this exercise: If $f\in L_p(\mathbb{R}^n)$ and $g\in L_q(\mathbb{R}^n)$, where $\frac{1}{p}+\frac{1}{q}=1$, Is their convolution $f\ast ...
1
vote
1answer
77 views

Showing a set of functions $F$ is bounded

I have a set of functions given by; $$F = \{f:[0,1]\rightarrow\mathbb{R}|\int_0^1 f(x)dx = 0, |f(x)-f(y)|\leq|x-y|, x,y\in[0,1]\}.$$ I have a solution for the question so my questions are about the ...
2
votes
4answers
109 views

Evaluating limit making it $\frac{\infty}{\infty}$ and using L'Hopital Rule

Let $P(x)=x^n+\displaystyle\sum\limits_{k=0}^{n-1}a_kx^k$. Find $$ \lim_{x \to +\infty} ([P(x)]^{1/n}-x) $$ I know that in order to solve this problem I need to multiply it by something that will ...
1
vote
1answer
135 views

Compact Subsets of $C[a,b]$

Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$ Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm. ...
0
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1answer
50 views

Prove the limit related to a recurrence

For the following sequence $\{a_n\}_{n=1}^{\infty}$, we define $a_1=\alpha\in(0,1)$, and for any $n\geq 2$, $a_{n+1}=a_n(1-a_n)$. Prove: $\lim_{n\rightarrow\infty}{na_n} = 1$.
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2answers
83 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
0
votes
1answer
513 views

Showing continuity using Weierstrass M test

Prove that $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}x^4$$ defines a continuous function on $\Bbb{R}$. My proof: For any $M>0$ and $x\in [-M,M]$, $$|\sum\frac{\cos(nx)}{n^2}x^4| \le \sum ...
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3answers
87 views

Finite lebesgue Integral

Hi guys I've been trying to prove this for a very long time, if someone could help me i would appreciated very much! let $(X,S,\mu)$ be a mesurable space, if $\mu(X)$ is finite and $f$ is a mesureble ...
0
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1answer
45 views

Raffle Odds and Payout

There are four raffles. One raffle ticket costs $\$500$. Raffle one sells $500$ tickets with winning ticket a payout of $\$80,000$. Second raffle one raffle ticket costs $\$150$ only sells $1,500$ ...
4
votes
1answer
173 views

Roots of $x^x-\tan (x)$

I conjecture, that the function $f(x)=x^x-\tan x$ has exactly one root in any of the intervals $\left[\dfrac{2n+1}{2}\pi,\dfrac{2n+3}{2}\pi\right]$ , where $n$ is a nonnegative integer. Does anyone ...
0
votes
1answer
54 views

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.Suggestion: take u to be the suitable cut off version of ...
2
votes
1answer
75 views

Laplacian on ${\bf R}^2$ and mean curvature

Consider a function $f$ on ${\bf R}^2$ whose critical point is origin. Then Gaussian curvature of graph of $f$ at origin is determinant of ${\rm Hess} \ f$ and Mean curvature is trace of ${\rm Hess} ...
2
votes
2answers
441 views

Limit goes to infinity, show that the f has a finite minimum.

So limit goes to infinity, and I have to show that there exists a finite infimum. how do i show this?
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0answers
99 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
0
votes
1answer
44 views

Prove: ∇⋅ϕF = ϕ∇⋅F + F⋅∇ϕ

I am asked to prove this identity using tensor notation. However, I am not sure where to even begin the problem.
0
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1answer
39 views

absolute and uniform convergence of a Fourier-like series

I am following stein's real analysis book and he claims that if $a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$ where $f\in L^1([-\pi,\pi])$ then $\sum_{n=-\infty}^{\infty} a_n r^{|n|}e^{inx}$ ...
2
votes
3answers
107 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
2
votes
2answers
75 views

$V(R)=Af(R\cdot B)$, where A and B are constant, prove that curl V is perpendicular to both A and B

If $V(R)$ can be expressed as $V(R)=Af(R\cdot B)$, where $A$ and $B$ are constant, prove that curl $V$ is perpendicular to both $A$ and $B$.
2
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1answer
49 views

For any open subset $A\subseteq \mathbb{R}$, $\operatorname{int}(\overline{A})=A$?

In the quiz of a class in MIT OCW, there is a T/F problem : For any open subset $A \subseteq \mathbb{R}$, $\operatorname{int}(\overline{A})=A$? The hompage of the class also provided a answer, and ...
1
vote
1answer
68 views

Is a closed n-dimensional disk compact necessarily compact?

As the title asks, is a closed n-dimensional disk compact necessarily compact? I'm thinking the answer would be no. If you consider the case in $\mathbb{R}^1$ then can you define the radius to be ...
3
votes
1answer
47 views

Evaluating limits by subsituting special sequences, justification for that

Sometimes I saw people using transformations like $$ \lim_{x\to 0} f(x) = \lim_{n\to \infty} f(\frac{1}{n}) $$ or $$ \lim_{x \to p} f(x) = \lim_{n \to \infty} f(x + \frac{1}{n}). \quad (*) $$ I know ...
0
votes
1answer
61 views

$a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
3
votes
3answers
2k views

Interior Points

I'm learning analysis using the Rudin's book, and sometimes the definitions make me wonder and leave me quite puzzled... So, interior points: a set is open if all the points in the set are interior ...
0
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1answer
20 views

Addition of distributions in statistics

Is it possible to add distributions? I've worked out "Say that you are given ten identical coins for which you assume Beta(4,4) prior distribution on the unknown probability θ of any of the coins ...
3
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0answers
46 views

inequality of some integrals of continuously differential function.

Let $f:[a,b]$→$\mathbb{R}$ be a continuously differential fuction satisfying f(a)=0. My goal is to show that $$\int_{a}^{b} |f(x)|^2 dx \le \frac{(b-a)^2}{2} \int_{a}^{b} |f'(x)|^2 dx $$ My ...
3
votes
1answer
118 views

Solution to a tricky inequality (math analysis)

Let $p>1$ and put $q=\frac{p}{p-1}$, so $1/p+1/q=1$. Show that for any $x>0$ and $y>0$, we have $$ xy \le \frac{x^p}{p}+\frac{y^q}{q}$$ And find where the equality holds. So far, I have ...
2
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1answer
31 views

how to show that $A(x)\nabla u\in L_\mathrm{loc}^{2}(\Omega) $ for $u\in H_\mathrm{loc}^{1}(\Omega)$

Let $\Omega\subset \mathbb{R}^n$ be a connected open set containing $0$, $u\in H_\mathrm{loc}^{1}(\Omega)$, $A(x)\leq C|x|^{-1+\epsilon}$, where $\epsilon$ is small, and we also have $$ \|\nabla ...
0
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1answer
33 views

Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?

Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$ converge to $f(x)$? I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
2
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0answers
99 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
0
votes
1answer
140 views

Integrals Regulated functions

stuck on an example for this question, Give an example of a regulated function $f \colon [a,b] \to \mathbb{R}$ with the properties that $\forall x \in [a,b] f(x) \ge 0 , \int_a^b f = 0$ and there is ...
0
votes
1answer
17 views

Using Taylor's series

Using Bayes Theorem I have solved a problem to the equation P(Dc|-) = (0.98-0.98p)/(0.98-0.93p) between the interval [0,0.1]. Show (e.g., by means of Taylor series) that in this interval the P(Dc|−) ...
2
votes
3answers
75 views

Minimum of set $\{\frac{m}{n} + \frac{4n}{m}\}$

We have the following set: $\mathcal{A} = \{ \frac{m}{n} + \frac{4n}{m};\ \ m, n \in \mathbb{N} \} $ Attempting to prove that the set's minimum is 4 yields: $$\frac{m}{n}+\frac{4n}{m} = \frac{m^2 + 4 ...
2
votes
1answer
149 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
2
votes
2answers
51 views

Show that $f: GL_{n}(\Bbb R) \to GL_n \Bbb (R): A \mapsto A^{-1}$ is infinitely differentiable

Consider the continuous image $f: GL_{n}(\Bbb R) \to GL_n \Bbb (R): A \mapsto A^{-1}$ I'm trying to proof with induction that $f$ is infinitely differentiable. I now understand how I can proof that ...
1
vote
4answers
149 views

How to prove that $a<S_n-[S_n]<b$ infinitely often

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that ...
1
vote
2answers
147 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
13
votes
3answers
591 views

Inequality involving partial sums of $\frac{|\sin{kx}|}{k}$

How to prove that $\forall x \in \mathbb{R}$, $n \in \mathbb{N}$, we have \begin{align} \sum_{k=1}^{n}\frac{|\sin{kx}|}{k}\ge |\sin{nx}| ? \end{align} I know that this partial sum will diverge ...
6
votes
3answers
259 views

How prove this nice limit $\lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$

let $f(x)$ is Continuous on $[0,1]$, and such $f(0)=f(1)$,and if $a$ is irrational number. show that $$\lim_{n\to+\infty}\dfrac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$$ where ...
1
vote
1answer
102 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
3
votes
2answers
473 views

Prove open set is not closed

The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY. Open set: A set $U$ is open if for every $a$ belonging to $U$, ...