Evaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$
I want to evaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$, where $g: \mathbb{R} \rightarrow \mathbb{R}$ is (Lebesgue)-integrable, and $f:\mathbb{R} \rightarrow \mathbb{R}$...
View Article$\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$
How to prove that $\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$ ?First try : I tried to compare the sum with the integral $\int_0^{+\infty} \frac{x^t}{1+tx}\mathrm{d}t$....
View ArticleConvergence of log-density ratio of KL divergence(discriminator) when...
In the paper "A Deep Generative Approach to Conditional Sampling", the author writes in the proof of Theorem 4.1:Since$$\Vert G^* - \bar{G}_\theta \Vert_{L^\infty(E_1)}\to 0, \quad \text{as } n \to...
View ArticleProving that a function $f$ converges to a number $L$ if and only if the...
This is an exercise that I am trying to prove. I am pretty sure that the proof of the first implication is correct. I am not so sure about the second one. I would appreciate if someone could check it...
View ArticleConvergence or divergence of $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$
If $a_1=a_2=1$ and $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$ How to prove convergence of the sequence and its limit or divergence?It is easy to see that the sequence is always positive and by that one can...
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Is the solution to...
Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration?I...
View ArticleClaim: Given ODE $\dot x = f(x)$, $f$ is locally lipschitz, then $x$ must be...
Can someone prove or disproof the claim: Given a locally lipschitz vector field $f$ with associated ODE $\dot x =f(x)$, then the solution $x$ must be locally lipschitzNote: local lipschitz...
View ArticleEvaluating Complex Series Convergence Radius by Hadamard's Theorem
Zorich's Mathematical Analysis Page 286:a) Making the formal substitution $z-a=(z-z_0)+(z_0-a)$ in the power series $\sum_{n=0}^{\infty}A_n(z-a)^n$ and gathering like terms, obtain a series...
View ArticleCan difference quotient sets be nowhere dense?
Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$Can $D_f$ be nowhere dense in...
View ArticleClosed form for $\int_0^{\frac{\pi}{2}}\left( \frac{1}{\log(\sin...
The integral has the numerical value$$\int_0^{\frac{\pi}{2}}\left( \frac{1}{\log(\sin x)}+\frac{1}{1-\sin x} \right)dx=0.86995763688\dots $$I have been unsuccesfully trying to find a closed form for...
View ArticleFind the $\inf_{f\in M}\sup_{x\in[0,1]}$ of the given expression
Let $M$ be a set of continuous decreasing functions $f$ on the segment $[0,1]$ for which $f(1)=0$. Find the$$\inf_{f\in M}\left(\sup_{x\in[0,1]}\frac{xf(x)}{\int_0^1f(t)\,\mathrm dt}\right)$$I will be...
View ArticleDetermine all Convergent Subsequences (with their limit) of the Sequence $1,...
Problem: Determine all convergent subsequences (with their limit) of the sequence $1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, \dots$I know that a subsequence of $\{a\}$ is a sequence...
View ArticleWhat is Laplace transform of $t^af(t)$ if Laplace transform of $f(x)$ is $F$.
What is the Laplace transform of $t^af(t),a>0$ if the Laplace transform of $f(x)$ is given to be $F(s)$. By definition, it should be like this:$$\mathcal{L}\{t^a f(t)\}(s) = \int_{0}^{\infty} t^a...
View ArticleIf $f$ is increasing real valued function and $g$ is defined as following...
Let $f:[a,b]\mapsto\mathbb{R}$ be increasing function. Now define, $g:[a,b]\mapsto\mathbb{R}$ such that $g(x)= f(x+)$ for all $x\in [a,b)$ and $g(b)=f(b)$. Now prove that $g$ is increasing function and...
View ArticleThe $L^p$ Norm of the Heat Kernel and its Divergence and Gradient
I am working with the Heat Kernel defined on Euclidean space with dimension $N$ as:$G(t,x) := \frac{1}{(4 \pi t)^{-N/2}} e^{-|x|^{2}/4t}$I have been told by my professor, and found on this post the...
View ArticleProving that a recursive sequence is bounded and monotone
Let $a_0=2\sqrt3$ and $b_0=3$, and define two sequences recursively by $$a_n=\frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}}\quad\text{and}\quad b_n=\sqrt{a_nb_{n-1}}.$$This is an exercise in Jay Cummings...
View ArticleWhich one of the following is TRUE(Real Analysis)? [closed]
Let $$f(x) =\begin{cases}x+1&x<0\\(x-1)^2&x≥0.\end{cases}$$Which one of the following is true?(a) f is differentiable on R(b) f has neither a local maximum nor a local minimum in R(c) f is...
View ArticleLipschitz Continuous Function in the Embedding Space
Given three metric spaces $(X,d_X),(Y,d_Y),(Z,d_Z)$ and an embedding function $g: X \rightarrow Z$, I am interested in finding more information about functions $f: X \rightarrow Y$ in which there...
View ArticleHow do I show $y\in B(x,\delta)\implies B(y,...
This is from "Supplement to Measure, Integration & Real Analysis" by Sheldon Axler. The definition of $B(x,\delta)$ is reproduced below.For $x\in\mathbb{R}^n$ and $\delta>0$, the open...
View ArticleDefining Sample Space for Stochastic Processes
I am studying stochastic processes and am struggling to understand how these processes can be properly defined.Consider the following experiment: I toss a coin, and if it lands on tails, I lose $1$. If...
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