MCQ: Non-negative series convergence question
Q: Let $\sum\limits_{n=0}^{\infty} a_n$ be a non-negative series. Which implication is true? [A] If the series $\sum\limits_{n=0}^{\infty} a_n$ is convergent, then $\sum\limits_{n=0}^{\infty} a_n^2$...
View ArticleIs the given property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded open sets such that $$ \lim_{m \to \infty} \sup \{r :...
View ArticleDoes $n(H_n-H_{n-n^k})\colon k\in[0,1)$ span the set $O(nH_n)-O(n)$
Given the following function, whose value depends on a function $g(n)$ of which we can only know its asymptotic growth:$$f(n)=\left(1-\frac{1}{n}\right)^{g(n)}$$I want to calculate its limit when...
View Articleshowing $y\to |y|^{p}$ is convex $p\geq 1$
$y\to |y|^{p}$ is convex only for $p\geq 1$ and $y\in \mathbb{R}$.This function is nondifferentiable but we can see that the second derivative is nonnegative in each interval $(-\infty,0)$ and...
View ArticleComplete monotonicity and $\sigma$-finiteness
Let $f:(0, \infty) \rightarrow (0, \infty)$ be a completely monotone function such that $f(0+)=\infty$, and by Bersntein Thoerem let $\mu$ be a positive measure on $[0, \infty)$ such...
View ArticleGeneralization of calculus that covers finite differences
Calculus stems from the concept of function and the derivative operator.In the case of non-standard calculus, the derivative operator is the forward differences operator with an infinitesimal step...
View ArticleIf we only specify one sequence of partitions in the definition of...
Update: This question differs from Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition? in the following way:From what I see...
View Articleproof of limits involving infinity using formal definition
I am reading my lecture notes on limits and stumbled across a 'solved' example problem.It concerns with the limit of $$ \lim_{x \to \infty} (x - \sqrt{x + 1}) = \infty$$Using the formal definition of $...
View ArticleHow do we prove that if $f(x)\leq g(x)$ in some deleted neighbourhood, then...
Suppose that\begin{equation*} \lim_{x \to a^-} f(x) = L \ \ and \ \ \lim_{x \to a^-} g(x) = M\end{equation*}If $f(x) \leq g(x) \ \forall x$ in a deleted neighborhood of $a$, then $L \leq M$.Proof....
View ArticleSolving inequalities involving nested modulus
Guys how do I solve inequality that has a nested modulus sign ? I'm familiar withsingle modulus, but I am quite lost here. This is an example of what I'm referring to, any help and explanation is...
View Articlecan certain sequances be Fourier coefficients
It is known that for each integrable function $f:[-\pi,\pi] \to \Bbb C$, with fourier coefficients $a_n$ ($n \in \Bbb Z$),$$ (1) \sum_{n=-\infty}^\infty |a_n|^2 <\infty$$ My question is This:Given a...
View Article$\lim_{x\to 0^+}\sum_{n=1}^\infty \frac{\chi(n)}{n} e^{-nx}$
In my answer to a question on this site I googled for a reference and found this. The answers of the user reuns of the latter linked question goes to great length to show that$$\lim_{x\to...
View ArticleProving the Independence of a Family of Trigonometric and Power Functions in...
Consider the family of functions $f_n$ and $(g_n)_{n\in\mathbb{N}}$, where for each $n$ in $\mathbb{N}$:$f_n: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f_n(x) = x^n \cos(x)$,$g_n: \mathbb{R}...
View ArticleContinuity Argument?
Let $f(x)$ be a nonnegative, nondecreasing and continuous function defined on $[0,a]$. Assume that if $f(x)\le \sqrt\epsilon$ then $f(x)\le \frac{1}{2} \sqrt{\epsilon}$ holds for sufficiently small...
View ArticleNeed examples of problems of the form "Construct a function $f:[0,1]...
Let $I$ be the unit interval. I developed a method to simplify problems of the form "Construct a function $f:I \rightarrow I$ satisfying given properties". These properties should only be expressible...
View ArticleFinding the interior of a set of continuous functions [closed]
Let $C[0,1]$ be the family of continuous functions on $[0,1]$ with thenorm $||x||_{\infty}=\max\limits_{0\leq t\leq 1}|x(t)|.$ Let$A=\{x\in C[0,1]| ||x||_{\infty}\leq 1, x(1)=1\}.$ Prove that$A$ is...
View ArticleProblem in mathematical analysis
Prove that the function $f(x) = x|\sin(x)|$ gets in the interval$(0,\infty)$ every positive value infinite times.In other words prove that for every $0<y$ the $f(x) = y$ equation have infinite...
View Articlecutoff function for fundamental solution
Consider the fundamental solution $h(x)=c_{n,s}|x|^{-(n-2)}$ defined for $x\in\mathbb{R}^{n}$, $x\neq 0$ with $c_{n,s}$ a normalizing constant. Is there some way to define this function so that it is...
View ArticleConvexity of product of two given functions
Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that...
View ArticleWhy Lebesgue measure? Why Borel σ-algebra?
Is any measure on any σ-algebra inside the power set of $\mathbb{R}^d$ a formal definition (or generalisation) of "volume" in $\mathbb{R}^d$?What's so special about Lebesgue measure that we choose it...
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