Given that $f\colon [-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$ What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so that I can make use […]

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-\epsilon) \left\| x-y\right\|$ for all $x,y \in \mathbb{R}^n$. Assume that for all $x \in \mathbb{R}^n$, the mapping $a \mapsto f_a(x)$ is continuous. Now let $x_0 \in \mathbb{R}^n$ be […]

I have a short question about partial integration. If I want to determine an integral of the form $\int f’gdx$, the formula for partial integration is: $$\int f’gdx=[fg]-\int fg’dx$$ https://en.wikipedia.org/wiki/Integration_by_parts . Sometimes it is useful to apply the integration rule twice, for example if $g=x^2$ and then you have to apply partial integration on $\int […]

In investigating of convergence of a sequence we use $n\longrightarrow \infty $ . why we can only use $\infty$ and we can not use the other numbers for convergence in a sequence as convergence of the other functions? thank you.

What is the relationship between singular points of algebraic curves (as described here or here), singular points of ode’s (as described here or here) and singular points in complex analysis (as described here or here)? They seem like three completely distinct ideas, yet I’d wager that the last two definitions fall out of the Taylor […]

Find $\displaystyle\lim_{n\to\infty}\sqrt{n}(\sqrt{n + a} – \sqrt{n}), a > 0$ Would it be right to start by rewriting it as $\displaystyle\lim_{n \to \infty} n^{1/2}((n+a)^{1/2} – n^{1/2})$?

Let $f \in C^{(n)}(-1,1)$ and $\sup_{-1 <x< 1}|f(x)|\leq 1$. Let $m_k(I) = \inf_{x \in I} |f^{(k)}(x)|$, where $I$ is an interval contained in $(-1,1)$. If $I$ is partitioned into three successive intervals $I_1,I_2,$ and $I_3$ and $\mu$ is the length of $I_2$, then $$m_k(I) \leq \frac 1\mu\left(m_{k-1}(I_1)+m_{k-1}(I_3)\right)$$ Things I have tried do not seem to […]

I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities: $\sum_{n=0}^\infty (-1)^n n!(n+2) = 1$ and $\sum_{n=0}^\infty (-1)^n n!(n^2-5) = -3$. Since no $p$ was stated, I figured there must be some trick that shows these are the values in any $\mathbb{Q}_p$, but I […]

Let $(f_n)$ be a sequence of continuous functions from the reals to the reals. And let $\sup{|f_n(x)-f_m(x)|} \leq \frac{1}{\min(m,n)}$ Prove there is a continuous functions $f$ such that $$\lim_{n\rightarrow \infty} \sup{|f_n(x)-f(x)|}=0$$ Start: I think this is an $\epsilon/3$ proof. Let $\epsilon >0$ and let $a$ be a real number. Since each $f_n$ is continuous, there […]

Suppose function $f:J→R$ on an open interval $J$ satisfies following inequality: $$\left|f(x)+f(y)−2f\Big(\frac{x+y}{2}\Big)\right|\leq C(f)\, |x−y|^{1+\delta}$$ for all $x,y∈J$, and for some $\delta \in (0,1].$ Is follow from this fact that $f\in C^{1}$ or $C^{1+\delta}$?

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