The sequence $(x_n)$ is defined as follows: $$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$ for all $n\geq 1$. a. Prove that $(x_n)$ has a finite limit and find that limit. b. For every $n\geq 1$, prove that $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

For each $n\in\mathbb N$, the function $f_n$ is defined on real numbers $x\ge n$ by $$f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.$$(a) If $n$ is fixed, prove that $\lim_{x\to+\infty}f_n(x)=0$. (b) Find the limit of $f_n(n)$ as $n\to+\infty$.

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as $$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$ Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent. [i]Ovidiu Țâțan[/i]

Let $f:(0,\infty)\to\mathbb R$ be a differentiable function. Assume that $$\lim_{x\to\infty}\left(f(x)+\frac{f'(x)}x\right)=0.$$Prove that $$\lim_{x\to\infty}f(x)=0.$$

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

consider $0<u<1$. find $\alpha > 0$ minimum such that there exists $\beta > 0$ satisfying $(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1$

Let $ H(n)$ be the number of simply connected subsets with $ n$ hexagons in an infinite hexagonal network. Also let $ P(n)$ be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh $ n$ in this hexagonal network. a) Prove that the limits \[ \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}\]exist. b) Prove the following inequalities: $ \sqrt2\leq\beta\leq2$ $ \alpha\leq 12.5$ $ \alpha\geq3.5$ $ \alpha\leq\beta^4$

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis. Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\] for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$ [b](i)[/b] Prove that $S$ is an infinite set. [b](ii)[/b] Find the least positive integer in $S.$ [b](iii)[/b] If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \] show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]

If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{no finite number}$

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

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 $\lim_{n \to \infty} x_n$.

Calculate the limit $$\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).$$ (For the calculation of the limit, the integral construction procedure can be followed).

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)