Prove that $\lim_{n\rightarrow\infty}\sqrt[n]{a_n}=L$
Suppose that $a_n>0$, $n\geq1$ and that $\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=L$. Prove $\displaystyle\lim_{n\rightarrow\infty}\sqrt[n]{a_n}=L$To resolve this problem, I solved...
View ArticleDifferentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
Use differentiation with respect to parameter obtaining a differential equation to solve$$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx$$No complex variables, only this approach. Interesting integral...
View ArticleEvaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$
What tools would you recommend me for evaluating this integral?$$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$$My first thought was to use the beta function, but it's hard to get...
View ArticleSolving $\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0$ for $M \in \mathbb{R}$
I am trying to solve the following nonlinear equation analytically:$$\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0 \, ,$$where $ M \in \mathbb{R} $ and $ 0 < \epsilon \ll 1 $.A solution can be expressed...
View ArticleContinuous function which satisfies the Luzin N property, but which does not...
My question is: can we find the function $g\in{\rm C}([a,b])$ which satisfies the Luzin N property on $[a,b]$, but which does not satisfy the Banach S property on [a,b]? Here $[a,b]$ is a compact...
View ArticleHow to prove the sum of limits theorem for a finite N number of limits?
I was reading a book with sequences and it proved that given two sequences $A$ and $B$ which both converge, then $\lim(A+B) =\lim(A)+\lim(B)$.However, the sum of $N$ limits...
View Articleproblem on double integral
Let $G:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be defined as$$G(t, x)=\begin{cases}t(1-x), & \text { if } t \leq x \leq 1 \\x(1-t), & \text { if } x \leq t \leq 1 \end{cases}.$$For a...
View ArticleProve that if $f$ is an element of Lip $\alpha$, then $f$ is uniformly...
I am required to prove that if $f$ is an element of Lip $\alpha$, then $f$ is uniformly continuous. Where Lip means a Lipschitz function. I also have to prove that if $f$ is an element of Lip $\alpha$...
View ArticleExercise 6.9 in Rudin's RCA (Real and Complex Analysis)
The following is an exercise 6.9 in Rudin's Real and Complex Analysis:Suppose that $\{ g_n \}$ is a sequence of positive continuous functions on $I=[0,1]$, that $\mu$ is a positive Borel measure on...
View ArticleCan a function be strongly differentiable but not continuously differentiable?
A similar question was asked before (however, there were a few issues with the definitions and answer given, as I pointed out over there): Can a function be differentiable but not strongly...
View ArticleOn the conditions of convergence in the generalized Riemann-Lebesgue lemma
The following generalizations of the Riemann-Lebesgue lemma are rather well known (see for example the paper Kahane, C. S., Generalizations of the Riemann-Lebesgue and Cantor-Lebesgue lemmas,...
View ArticleWhat is the Haar measure on the unit sphere?
I try to understand the proof of Lemma 4.2. in the paper 'The Euler equations as a differential inclusion' by De Lellis and Székelyhidi. In the proof they use the Haar measure on the unit sphere...
View ArticleDifferentiability implies continuity of $f(x,y)$
I have seen proofs of this statement on the site however none use the definition of differentiability I am familiar with which is why I'm asking this question. My proof is as follows:If $f(x,y)$ is...
View ArticleConditions on coefficients of double convergent series for it to be...
I am working on an exercise from a graduate first year complex analysis class. The problem states:let $f: D \to \mathbb{C}$ given by $ f = u + iv$. Assume that $f$ admits an absolutely convergent...
View ArticleA problem on finding the limit of the sum
$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$Show that, $\lim_{n\rightarrow\infty} u_n = 0$.The only approach I can see is either finding...
View Articleconvex set and $L^2[a,b]$
Am I right if I say $X=\{f\in L^2[a,b] \mid \lVert f\rVert_{L^2} \le 1\}$ is a convex space?Let $f,g\in L^2[a,b]$ and let $\theta\in[0,1]$. $\lVert \theta...
View ArticleSmall Question from the Proof of Poisson Summation
Let $f$ be a Schwartz function on $\mathbb R$. I want to prove that$$\sum_{n \in \mathbb Z}f(n) = \sum_{n \in \mathbb Z}\widehat{f}(n)$$The beginning of my proof goes like this: the decay property of...
View ArticleThe product of a Dedekind cut and its inverse equals one
Exercise 1.11 of the textbook Real Mathematical Analysis by Pugh asks what the inverse of a cut $x=A|B$ is ($x$ being positive).The first step is to show that $x^*=C|D$ where $C=\{1/b: b\in B...
View ArticleProof of factor theorem regarding polynomials
Division of polynomials: Let $f$ and $g$ be polynomials with $g(x)\neq 0$. Then there exist unique polynomials $q$ and $r$ such that$r=0$ or $\deg r<\deg g$, and$f(x)=q(x)g(x)+r(x)$.I want to prove...
View ArticleConvergence in $L^p$ norm implies pointwise convergence almost everywhere?...
Fix a real number $1\leq p<\infty$. Is it true that if functions $f\in L^p(\mathbb{R})$ and $f_1,f_2,\ldots\in L^p(\mathbb{R})$ are such that $\|f_n-f\|_p\rightarrow 0$ as $n\rightarrow \infty$,...
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