$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$. [i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

$f: (0,1) \rightarrow [0, \infty)$ is zero except at a countable set of points $a_{1}, a_2, a_3, ... $ . Let $b_n = f(a_n)$. Show that if $\sum b_{n}$ converges, then $f$ is differentiable at at least one point. Show that for any sequence $b_{n}$ of non-negative reals with $\sum b_{n} =\infty$ , we can find a sequence $a_{n}$ such that the function $f$ defined as above is nowhere differentiable.

(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$. (b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Prove that $$\int_0^{\pi/2}f(\sin(2x))\sin x\, dx = \int_0^{\pi/2} f(\cos^2 x)\cos x\, dx.$$

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote \[ S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|. \] a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ \sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent? b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

Let $K$ be a closed subset of the closed unit ball in $\mathbb{R}^3$. Suppose there exists a family of chords $\Omega$ of the unit sphere $S^2$, with the following property: for every $X,Y\in S^2$, there exist $X',Y'\in S^2$, as close to $X$ and $Y$ correspondingly, as we want, such that $X'Y'\in \Omega$ and $X'Y'$ is disjoint from $K$. Verify that there exists a set $H\subset S^2$, such that $H$ is dense in the unit sphere $S^2$, and the chords connecting any two points of $H$ are disjoint from $K$. EDIT: The statement fixed. See post #4

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ [b]a)[/b] Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ [b]b)[/b] Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$ [i]A. Hajnal[/i]