Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.

a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$ to be this limit. b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$

Let be a real number $ a\in (0,1) $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that: $$ \lim_{x\to 0} f(x) =0= \lim_{x\to 0} \frac{f(x)-f(ax)}{x} $$ Prove that $ \lim_{x\to\infty } \frac{f(x)}{x} =0. $

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

a) Evaluate \[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\] with $a,b>0$. b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and \[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\] Prove that: 1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded. 2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent. [i]Valentin Matrosenco[/i]

Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.

[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded. [b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that $$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$ for any natural number $ n. $ Prove that the following relations are equivalent: $\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $ $ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $ [i]Eugen Păltănea[/i]

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $ 247,$ $ 275,$ and $ 756$ and end with the term $ 824.$ Let $ \mathcal{S}$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $ \mathcal{S}?$ $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

For each $ c\in\mathbb C$, let $ f_c(z,0)\equal{}z$, and $ f_c(z,n)\equal{}f_c(z,n\minus{}1)^2\plus{}c$ for $ n\geq1$. a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded. b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.

Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence $$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$ tends to $ 2. $ [i]Mihai Bălună[/i]

Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

Let $ a$ be a positive real number, in Euclidean space, consider the two disks: $ D_1\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}a\}$, $ D_2\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}\minus{}a\}$. Let $ D_1$ overlap to $ D_2$ by rotating $ D_1$ about the $ y$ axis by $ 180^\circ$. Note that the rotational direction is supposed to be the direction such that we would lean the postive part of the $ z$ axis to into the direction of the postive part of $ x$ axis. Let denote $ E$ the part in which $ D_1$ passes while the rotation, let denote $ V(a)$ the volume of $ E$ and let $ W(a)$ be the volume of common part of $ E$ and $ \{(x,\ y,\ z)|x\geq 0\}$. (1) Find $ W(a)$. (2) Find $ \lim_{a\rightarrow \infty} V(a)$.

For $a>0,$ find $\lim_{a\to\infty}a^{-\left(\frac{3}{2}+n\right) }\int_{0}^{a}x^{n}\sqrt{1+x}\ dx\ (n=1,\ 2,\ \cdots).$

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.