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 ArticleCalculus: Is $(x-1)/f(x)$ decreasing?
Suppose $f$ is a continuous function and $f(x) \geq x-1$ for all $x>1$. It is also given that $f(x)$ is a monotonically increasing function in $x>1$. Can I say that$$\frac{x-1}{f(x)}\quad\text{is...
View ArticleWhy is log-of-sum-of-exponentials $f(x)=\log\left(\sum_{i=1}^n e^...
How to prove $f(x)=\log\left(\displaystyle\sum_{i=1}^n e^{x_i}\right)$ is a convex function?EDIT1: for above function $f(x)$, following inequalities hold:$$\max\{x_1,x_2,\ldots,x_n\}\leqslant...
View ArticleCan an unbounded strictly increasing sequence be convergent? [duplicate]
I was doing excercises on convergence of sequences of real analysis but I came up with a problem I don't know how to prove.Note: The book has not shown Cauchy sequences yet and I don't know what they...
View ArticleSolution of the "reciprocal of the heat equation"? [closed]
I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$...
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Rudin's RCA 2.24 theorem : Lusin's theorem
I have a question concerning the affirmation : "There are compact sets $K_n$ and open sets $V_n$ such that $K_n \subset T_n \subset V_n \subset V$ and $µ(V_n-K_n)<2^{-n} \epsilon$."How do we know...
View ArticleFractional regularity and the quotient $\frac{f(x)-f(y)}{|x-y|^\alpha}$
Holder, Sobolev, and Besov spaces are often used to measure regularity, and in particular, fractional regularity. On one hand, the relation between their respective norms and...
View ArticleWhy is a lot of Fourier analysis done on an annulus?
I am studying harmonic analysis from these lecture notes and a lot of results and definitions always assume that the Fourier transform of a function has support in an annulus or a ball. The same...
View ArticleDerivative of a distance function
I have a question about a derivative of a distance function.Let $D$ be a bounded and connected open subset of $\mathbb{R}^{d}$ with Lipschitz boundary. We define the following distance function $F$ on...
View ArticleA very interesting formula for $\sin x$
Prove that for all $(n,t,x) \in \mathbb{N}^{*}\times \mathbb{R}\times]0,\pi[$ : $\sin(x) = \left( \sum_{k=1}^{n}t^{k-1}\sin kx \right)\left( 1-2t\cos x+t^{2} \right)+t^{n}\left( \sin((n+1)x)-t\sin(nx)...
View ArticleIf a sequence of functions is zero almost everywhere and converges pointwise...
Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $\mathcal{L}^p(\mathbb{R})$. Each $f_n$ is zero almost everywhere. Additionally, the sequence converges pointwise almost everywhere to some...
View ArticleDo three points of inequality between convex functions imply inequality over...
Say I have 2 convex functions, $f:\mathbb{R}\rightarrow\mathbb{R}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$. I want to prove that $f(x) > g(x), \forall x \in [l, u]$.I know that $f(l) > g(l)$ and...
View ArticleWhat is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n...
Let $(T_n)_{n>0}$ be an increasing sequence of positive integers and $c>0$ a positive number such that $\lim_{n\to\infty}\frac{T_n}{n}=c$. Write $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ for a real...
View ArticleGiven a functional equality prove that the function is never $0$
For a function $u:\mathbb{R}\to\mathbb{R}$ with $u(t)^2+u(t)=t^2+t+2$ , $u(0)=1$, Can we prove that the graph of $u$ never touches the $t$ axis ?We can easly see that by adding $0.25$ to both sides of...
View ArticleApproximating powers of elements on the unit circle
Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if:$$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| <...
View ArticleEstimation of Sum of Binomial Coefficients that doesn't Start from Zero
I am having some trouble showing the following estimation:Let $\mu \in (0, 1)$ be an absolute constant. There exists an absolute constant $\eta \in (0, 1)$ small enough such that the number of subsets...
View ArticleProving that the set of limit points of a set is closed
From Rudin's Principles of Mathematical Analysis (Chapter 2, Exercise 6)Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed.I think I got it but my argument is a bit hand...
View ArticleAn integral inequality involving the Bernoulli polynomials
The classical Bernoulli polynomials $B_j(t)$ are generated by\begin{equation*}\frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad...
View ArticleThe convergence of this series: $\sum\limits_{n=2}^\infty {1\over n^{\log n}}$
I came across a problem on convergence of series and I did not get into any idea about this -- any help or hints ? $$\sum_{n=2}^\infty {1\over n^{\log n}}$$ What about $n^{\log(\log n)}$ ?
View ArticleNonstandard Analysis research project ideas [closed]
Before I ask my question, this is my first post on MSE, so I apologise if I have not met the standard etiquette of a post.I have finished my first year at a UK univeristy for a maths degree. In our...
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