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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1
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0answers
11 views

Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ ...
4
votes
4answers
300 views

Is it possible / allowed to use L'Hôpitals rule for products?

In our readings, we had L'Hôpitals rule and defined it like that: $\lim_{x\rightarrow x_{0}}\frac{f'(x)}{g'(x)}$ Because we had it in our readings, we are allowed to use this to find limit of ...
3
votes
4answers
68 views

Analyze if this sequence converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$

Analyze if this sequence converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$ I have used ratio test: $\lim_{n\rightarrow \infty}\left |\frac{a_{n+1}}{a_{n}} \right |< 1$ $\Rightarrow$ $\lim_{n\...
2
votes
2answers
79 views

Very strange - what's the limit of $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$?

What's the limit of: $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$ ? $\lim_{x \rightarrow 0} \left (sin(x) + cos(x) \right) = sin(0)+cos(0) = 1 $ $\lim_{x \rightarrow 0} x = 0$ $\Rightarrow \frac{...
5
votes
6answers
97 views

Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $

How can I prove that $${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $$ I tried the following: We use the falling factorial power: $$y^{\underline k}=\underbrace{y(y-1)(...
4
votes
1answer
29 views

Definition second differential of a vector field

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function. Then we know that its differential $df: \mathbb{R}^2 \rightarrow Hom(\mathbb{R}^2,\mathbb{R}^2)$ maps vectors to matrices/linear ...
2
votes
2answers
27 views

Show that the following equation has got exactly one solution for each $C>0$

Show that the equation $$C=\left ( 1+x+\frac{1}{2}x^{2} \right)*e^{-x}$$ has got exactly one solution for each $C>0$. Alright so I did it like that but not sure if it's correct: $0<\left ...
3
votes
2answers
457 views

Proof that the following function is a polynomial

I've been trying to get my head around this problem for a long time, yet I have not been able to make much progress. Let $\ell_0(j) = \left\lfloor \frac{1}{2}\left( \sqrt{8j^2 - 8j + 1} + 2j - 1 \...
5
votes
1answer
31 views

Limit of sequence $\lim_{n\to\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$

This is no homework. It's another task of a sample exam and I'd like to know how to solve it. Find the limit of $$\lim_{n\to \infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$$ Both ...
31
votes
3answers
660 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, H_k^{(1)}\...
7
votes
1answer
1k views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and $...
1
vote
1answer
30 views

If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(x)$ is 2, prove that the image of $f$ is an open set.

I don't see how to solve the following problem, any suggestions? Let $f:\mathbb{R}^{3}\to \mathbb{R}^{2}$ such that $f\in C^{1}$. If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(...
4
votes
1answer
972 views

Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$

The question is as follows: Given: (1) function $f: U \subset \mathbb R^n ==> \mathbb R$ (2) $U$ is open and convex set (3) $f \in C^1$ in $U$ Goal: Show that $f$ is ...
3
votes
1answer
39 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...
6
votes
3answers
289 views

Limit of the sequence $\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$, strange result

$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$ $\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \...
-1
votes
1answer
32 views

help with real analysis [on hold]

Let $S \subset \mathbb{R}$ be nonempty. Show that if $u= \sup S$, then for every number $n$ belong to $\mathbb{N}$ the number $u -\frac{1}{n}$ is not an upper bound of $S$, but the number $u + \frac{1}...
0
votes
2answers
23 views

Differential $\mathrm{d}^2f$ of implicit function $F(x,y,z)=xyz-x-y-z=0$

Determine the differential $\mathrm{d}^2f$ of the implicit function defined as $z=f(x,y)$: $$F(x,y,z)=xyz-x-y-z=0$$ So in fact of the implicit function I have to use the implicit function ...
1
vote
2answers
44 views

I find correct limit of the sin cos function?

This is no homeworks I only do for learn. $$\lim_{x\rightarrow \pi}\frac{\sin^{2}x}{1+\cos x}$$ I use l'Hôpital rule because no idea where limit go for both. Top is called $g(x)$ and bottom is $...
0
votes
2answers
19 views

How would I use the difference quotient on this logarithmic function?

This is no homework, it's for exam practice. Show that $\lim_{x\rightarrow 0}\frac{1}{x}ln(1+ax) = a$ where $a \in \mathbb{R}\setminus \left \{ 0 \right \}$ is chosen definitely / fixed (...
1
vote
1answer
21 views

Need know all ways to show function is continuous, convergent and differentiable [on hold]

Please tell me all ways to show / proof that a function is continuous, convergent and differentiable. continuous: show that function is differentiable if yes then it is continuous also convergent: ...
1
vote
2answers
31 views

Limit of functions - always for both sides (+-) necessary?

I'm very confused when I read some pages on the internet about limits (for functions). Let's say I got any function f(x) given and someone tells me to find the limit (towards 3 or $\infty$ or ...
3
votes
2answers
48 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n+...
1
vote
1answer
393 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
0
votes
1answer
33 views

Upper hemicontinuity of a correspondence

I would like to know whether the following correspondence is upper hemicontinuous: $$ C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases}, $$ ...
4
votes
1answer
54 views

Integral inequality with gradient

Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
-3
votes
0answers
27 views

Using Rolle's Theorem to prove that f'(c)=rf(c). [on hold]

Suppose that the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)=0$. Prove that for each real number $r$, there is some $c$ on $(a,b)$ such that $f'(c)=r\cdot f(c)$. ...
2
votes
5answers
143 views

Definite integral of $\sqrt{\frac{1}{\cos^2(x)}}$

I've got problems with this integral: $$\int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2(x)}} \, \mathrm{d}x$$ First I substitute $x=2\arctan(x)$ but this leads nowhere. Any hints for solving?
1
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0answers
24 views

How to solve this Sturm Liouville problem?

$\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$ Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
0
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0answers
15 views

Non integer, non-centered Gaussian moments

I have read the following question : Non-centered Gaussian moments where it is stated that : $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{...
17
votes
2answers
432 views

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

I was trying to find closed form generalizations of the following well known hyperbolic secant sum $$ \sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n}=\frac{\left\{\Gamma\left(\frac{1}{4}\right)\right\}^2}{...
0
votes
1answer
52 views

Requirements for the Functional Equation $\sum_{i=1}^{n}f_i(x)g_i(y)=0$

Consider the following functional equation $$\sum_{i=1}^{n}f_i(x)g_i(y)=f_1(x)g_1(y)+f_2(x)g_2(y)+\cdot\cdot\cdot+f_n(x)g_n(y)=0 \tag{1}$$ where $f_i(x)$ and $g_i(y)$ are arbitrary functions. ...
0
votes
0answers
27 views

How to draw conclusion about Limits

Let $f$ and $g$ be two $\mathbb{R}\to\mathbb{R}$ functions such that $$ \limsup_{t\to\infty}\frac{1}{t}\log\left(\left|f(t)\right|\right)>\limsup_{t\to\infty}\frac{1}{t}\log\left(\left|g(t)\right|\...
6
votes
2answers
169 views

When adding zero really counts …

Note: Although adding zero has usually no effect, there is sometimes a situation where it is the essence of a calculation which drives the development into a surprisingly fruitful direction. Here is ...
5
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2answers
58 views

Finding a special subsequence of any Cauchy sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Let $(\epsilon_n)$ be a sequence of real numbers and decrease to $0$. Show that there is a subsequence $(x_{n_k})$ of $(x_n)$ ...
2
votes
1answer
105 views

Compute the sum of the series.

I just see the equality in my textbook, but I really have no idea how it arises (maybe it is obvious to the author), and it seems Fourier methods are not applicable. I would appreciate if someone ...
1
vote
1answer
28 views

Why is compactness required for Brunn-Minkowski theorem?

Brunn-Minkowski theorem reads as follows: Consider two nonempty compact sets $A, B \subset \mathbb{R}^n$. Then the following inequality holds $$ [M(A+B)]^{\frac{1}{n}} \geq [M(A)]^{\frac{1}{n}} + [M(...
1
vote
1answer
21 views

particle velocity variable with time [on hold]

Consider two fixed points $A$ and $B$. At $A$, you have a receiver and at $B$ you have a transmitter. $B$ continually emits particles towards $B$ at a constant rate. But the particle velocity is ...
0
votes
1answer
54 views

Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
0
votes
1answer
38 views

Analyze the monotony of this function - I almost got it

Analyze the monotony of $f(x) = (1+x+\frac{1}{2}x^{2})e^{-x}$ To analyze the monotony of a function, you can build the first derivation of the function, equalize it with $0$ and if the (derivative) ...
0
votes
0answers
17 views

Equality case in the Prékopa-Leindler inequality.

in the paper 'Remarks on the conjectured log-Brunn-Minkowski inequality' by C. Saraoglou, the author uses the result (Lemma A. 3.) about the equality case in the Prékopa-Leindler inequality. For the ...
2
votes
4answers
164 views

Value of tan(pi/2) [duplicate]

I understand that this is a very stupid question but I'm not getting the answer. At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$? Text tells it to be $+\infty$. But ...
0
votes
1answer
16 views

Multivariable implicit function - Jacobi Matrix

Find the derivate $f',f''$ of the implicit function $z=f(x,y)$ defined by the following equation: $$F(x,y,z)=x^2+y^2+z^2-a^2=0$$ So the first step to build the Jacobi-Matrix $f'$ lead me to ...
3
votes
1answer
70 views

Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...
435
votes
19answers
42k views
2
votes
1answer
63 views

Are derivatives actually bounded?

Suppose you a function $f$ which is differentiable, with the property that $$ f^{(n)} (0) = (n!)^2 $$ And in general $$ f^{(n)} (a) = O((n!)^2)$$ For any $a \in \mathbb{R}$. This function ...
0
votes
1answer
31 views

A question about Inverse Function theorem

The Inverse Function theorem: Let A be open in $R^n$; let $f: \to R^n$ be of class $C^r$. If $Df(x)$ is non-singular at the point $a$ of $A$, there is a neighborhood $U$ of the point $a$ such that $f$ ...
3
votes
0answers
52 views

Show the equality

Let $(X, \mathcal{A}, \mu)$ space with measure, $\mu(X) = 1$, $\epsilon > 0$ and $f: X \rightarrow [\epsilon,\infty)$ a $\mathcal{A}$-measurable and bounded function, I've tried show $$\lim_{p \...
1
vote
1answer
35 views

Necessary condition for convergence of Riemann integrals

For integrals of the form $\int_0^{\infty} |f(x)| dx$, is it necessary that $lim_{x \rightarrow \infty} f(x)$ = 0 for the integral to converge?
0
votes
0answers
24 views

I want to show that the function space $C_0(X)$ is Banach [duplicate]

I'm reading some papers but I encountered a problem that "$C_0(X)$ is Banach space". Here $$ C_0(X):= \{ f: X\to \mathbb{C}: f \text{ is continuous and } \forall \epsilon>0, \exists K(\text{compact}...
0
votes
2answers
44 views

Proving $f(x)=x^2 +3x+1$ is continuous on $(0,1)$ by definition

Def: A function is continuous if f : I → R at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I, |x-c| < δ => |f(x) - f(c)| < ε Prove that $f(x)=x^2 +3x+1$ is ...