What is the easiest or shortest way to show $A = \sum_{n=1}^{\infty}...
What is the easiest or shortest way to show$$ A = \sum_{n=1}^{\infty} \frac{1}{n(n+6)(n+7)} = \frac{223}{5880} $$?Now A has many forms$$A = \sum_{n=1}^{\infty} \frac{1}{n(n+6)(n+7)} =...
View ArticleInfinitely differentiable function with given zero set and given one set
Consider $V\subseteq U\subseteq\mathbb{R}^n$, where $V$ and $U$ are both open sets and $\partial V\subset U$.Is it possible to construct a $\mathcal C^\infty$ function...
View Articleproving a function is integrable using polar coordinates
Prove that the function $$ f(x, y) = \frac{(x - y) \sin(xy)}{(x^2 + y^2)^2}$$ isintegrable on $[-1, 1] \times [-1, 1]$.My idea goes like thisLet $x=r\cos\theta, y=r\sin\theta$We have $|f|\le...
View ArticleHelp justifying this formula from my physics class
I was doing a physics class and the following analysis was done. It is about first order approximation of a function. The function is given by:$$f(x,y) = 1 +...
View ArticleMeasurablity of summands
Assume that $(X, \mathcal{R}, \mu)$ and $(X, \mathcal{R}, \nu)$ are premeasures on a ring $\mathcal{R}$ such that there exists $\{A_n\} \subset \mathcal{R}$ where $A_n \uparrow X$ and $\mu(A_n) <...
View ArticleHighlevel Characterization of sets with lebesgue measure equal to 0 [closed]
I have a theory on why lebesgue measure is non zero for crazy sets like cantor sets which are not made of intervals.Let $S$ be the set for which you want to find measure.Define : For $x \in S$, $f(x) =...
View ArticleCounter example of Lebesgue dominated convergence theorem if $f_n$ is not...
Let $f$ and $f_n$ be nonnegative measurable function on $[0,1]$ such that $f_n\to f$ pointwise, and we may assume that $f_n\leq f$ for all $n$ in $\mathbb{N}$.How to find a counterexample to show that...
View ArticleFind a sequence of real-valued nonnegative functions $f_n$ on $[0,1]$ such...
Context: I am looking for a sequence of functions $(f_n)$ on $[0,1]$ such that $\lim \sup_{n\to\infty}f_n(x)=\infty$ for all $x\in[0,1]$ and $$\lim_{n\to\infty}\int_0^1f_n(x)\text{d}x=0$$Attempt: I'm...
View ArticleLog function Big-Oh
I need to show that $\log(1+5\epsilon) = O(\epsilon)$ as $\epsilon \rightarrow 0$ from the right hand side.I believe I can do this using Taylor's theorem or L'Hopitals rule.If the original statement...
View ArticleMust continuous $H^1(\mathbb{R}^2)$ function tend to zero at infinity?
Here, $H^1(\mathbb{R}^2)$ is the standard Sobolev spaces for $L^2(\mathbb{R}^2)$ functions whose weak derivative belongs to $L^2(\mathbb{R}^2).$My question in the title comes from calculus of...
View Articleshow $[0,1]$ is uncountable using outer measure
This is a question from Real Analysis by Royden, 4th edition. (#5, pg. 34)Using properties of outer measure, prove that $[0,1]$ is uncountable.I believe that I am going to have to assume otherwise and...
View ArticleUsing the $\varepsilon$-$\delta$ definition of a limit, show that $\lim_{x\to...
Using the $\varepsilon$-$\delta$ definition of a limit, show that$$\lim_{x\to2}\frac{x^3+2x^2-8x}{x-2}=12$$My first thought it to remove the singularity, by factorising out $(x-2)$ and...
View ArticleHow can the improper integral $\int_0^\infty \sin(t^2)dt$ converge? [duplicate]
According both to Wolfram Alpha and to sources like Wikipedia, the improper integral $\int_0^\infty \sin(t^2)dt$ converges, but this seems counterintuitive or even impossible. The sine function...
View ArticleIs a Pi system comprised of independent events always a lambda system?
If you had a pi-system such that all events are independent, will this always result in a lambda system (so then the pi-lambda theorem can be applied)?
View ArticleSet of Subsequential limits of a sequence is connected
Let $x_n$ be a bounded sequence in $\mathbb{R}^n$ such that $\sum_{k\in \mathbb{N}} \| x_{k+1} - x_{k}\|^2 < +\infty$.Let $S$ be the set of all subsequential limits of $x_n$, that is:$$S =...
View ArticleInfinite Product...
I've been looking at proofs ofEuler's Sine Expansion, that is$$\frac{\sin\left(x\right)}{x}=\prod_{k = 1}^{\infty}\left(1-\frac{x^{2}}{k^{2}\pi^{2}}\right)$$All the proofs seem to rely on Complex...
View ArticleQuestion on continuity of Stieltjes premeasures in prooving continuous at...
The citations are just for reference. The question here is not rooted in some details in this proof but in the intuition behind the Stieltjes measures.(3.3.1) $\mathcal{A}$ is dyadic half-open...
View ArticleProving $\lim_{x\to 10} \ln(x - 3) \neq \infty$ by epsilon-delta
I am not good at those proofs, and I cannot find any resource because the part of limits is always reduced to the minimum, with few explanations.$$\lim_{x\to 10} \ln(x - 3) \;=\; +\infty$$This is...
View ArticleDensity Theorem and Limit Proof
I'm new to proof writing and have trouble understanding hoe to structure or get proofs started. An example proof that my teacher left as practice is below. Any suggestions? Of course, I know I must use...
View Article$f(x) + f(-x) = f(x)f(-x)$ for $f : \mathbb{R} \rightarrow \mathbb{R}$ [closed]
I could not find any information on the internet relating to this problem. I am assuming that the solution is very broad but not all functions work. What are the solutions of the functional equation?
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