How to solve (or verify) the following inequality?
The following question is directly related to this one and involves the resolution (or verification) of an inequality. Again, the first part of my question is dedicated to the creation of the...
View ArticleComparison test to determine the convergence of the integral...
QuestionI am looking for a comparison to test the convergence of the integral.$$\int_0^{\frac{\pi}2} \frac{1}{\cos(x)^{0.5}} \mathrm dx$$AttemptI can see that this integral should be bounded above by...
View ArticleProve that $f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$ is continuous almost...
Let $f:(a,b) \rightarrow \mathbb{R}$ be a bounded measurable function, and $f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$. Take a sequence of function $\{j_n\}$,$j_n \in C^\infty$, $j_n \ge 0$, supp $j_n...
View ArticleStieltjes Integrable and series
I have the following conjecture that I would like to see proven or disproven:Let $\{\alpha_i\}$ be a sequence of monotonically increasing functions on $[a,b]$. Suppose $f\in \mathscr{R}(\alpha_i),...
View ArticleProve that $ f(x)=\sum\limits_{n=3}^{\infty} \frac{\min\limits_{k\in Z} |4^n...
I have this exercise from elementary analysis:Define the function $f: \mathbb R \to\mathbb R$ by$$ f(x)=\sum\limits_{n=3}^{\infty} \frac{\min\limits_{k\in Z} |4^n \cdot x - k|}{4^n} $$Prove that(1) $f$...
View ArticleProve that $f_n(x) ā f(x)$ uniformly on $E$ as $n āā.$
Let $E ā \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that...
View ArticleUniform Convergence Given Monotone Sequence
I've been working on this problem for a few days and it is a bit frustrating, especially because I know the solution is probably right in front of me.Problem. Suppose $f_n:[a,b]\to\mathbb{R}$ is a...
View ArticleAsymptotic growth of a set of functions
I need to test a condition on a function that must grow faster than $\Theta(n^2)$ but slower than $\Theta(n^2H_{n^2})$ when $n\to\infty$($H_{n}$ is the Harmonic Number, so it can be replaced by its...
View ArticleUniform convergence of continuous functions $ \{f_n\} $ on $(a,b)$
I didn't add a lot of context so I just edited it.If I have a sequence of functions $ \{f_n\} $ which are continuous on $[a, b]$, and if they converge uniformly on every subset of $(a, b)$: $\forall S...
View ArticleHow to calculate the diameter of a set in $\ell^{2}$
He had studied functional analysis in the book "Lectures on Functional Analysis and the Lebesgue Integral" by V. Komorkik. On page 9 there is an example of properties in finite-dimensional spaces that...
View Articlehow to find the lagrange form of remainder for the expansion of arctan x.
The precise task was to determine the finite series of $\arctan x$ with the Lagrange form of the remainder.While going through the process, I tried to find any pattern in the successive derivatives but...
View ArticleShow $\tan(x)-x>0$ ,$\forall x \in (0,\pi/2)$
I know the derivative is greater than $0$ for all $x$ in $(0, \pi/2)$, but how to show $\tan(x)-x $ is greater than $0$ as $x$ approaches $0$? Note: we have not yet learned l'hospitals rule.
View ArticleProb. 26, Chap. 5 in Baby Rudin: If $\left| f^\prime(x) \right| \leq A \left|...
Here is Prob. 26, Chap. 5, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f$ is differentiable on $[a, b]$, $f(a) = 0$, and there is a real number $A$ such that...
View ArticleKernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $x_i$ are...
View ArticleFind $\lim_{n\to \infty }\frac{\left(2n^{1/n}-1\right)^n}{n^2}$ [duplicate]
Find $\lim_{n\to \infty }\left(2n^{1/n}-1\right)^n/n^2$.I can use L'Hospital rule and differentiate it two times. But is there a simple way to solve this?
View Article$f$ integrable implies the existence of a continuous $g(x) \le f(x)$ where...
Prove that if $f:[a,b]\to\mathbb{R}$ is integrable, $\forall\epsilon\gt0, \exists g:[a,b]\to\mathbb{R}$ such that g is continuous, $g(x)\leq f(x)$$\forall x\in[a,b]$, and...
View ArticleContinuity of optimizers of the Hopf--Lax formula
Let $f:\mathbb{R}^d\to \mathbb{R}$ be smooth and convex with at most linear growth (you can assume Lipschitzness if it simplifies). Fix any $t>0$. It is clear that a maximizer of the following...
View ArticleHolder Inequality Generalized? Is it true that...
I am working on a certain problem and a way to solve it would be to prove an inequality similar to the Holder inequality for sums. The inequality is:Let $p,q>1$ and $a,b\geq1$ such...
View ArticleProve or disprove $\sum\limits_{\mathrm{cyc}} \sqrt{a_1a_2} \ge...
Problem. Let $n\in \mathbb{N}_{\ge 3}$.Let $0 \le a_1 \le a_2 \le \cdots \le a_n$. Prove or disprove that$$\sqrt{a_1a_2} + \sqrt{a_2a_3} + \ldots + \sqrt{a_na_1} \ge \sqrt[3]{a_1a_2a_3} +...
View ArticleBooks on mathematical constructions
Iām looking for suggestions and recommendations for books that focus on constructions in mathematics, or at least have exercises which focus on construction-based proofs.In particular, I seem to...
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