Found problems: 837
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
Let $f$ and $g$ be two functions such that
\[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\]
Find the domains of $f$ and $g$ and then prove that
\[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]
Show that $ \pi$ is irrational.
Are there positive integers $a, b, c$ greater than $2011$ such that:
$(a+ \sqrt{b})^c=...2010,2011...$?
Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that
\[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\]
holds for all real numbers $x,y$.
Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by:
$ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$
contains infinitely many natural numbers.
Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection of the positive integers. Prove that at least one of the following limits is true: \[\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.\][i]Proposed by Dylan Toh[/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$
If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$
$$(2-a_n )a_{n+1} =1,$$
prove that $\lim_{n\to \infty} a_n =1.$
Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties:
$(\text i)$ $|x_n|$ is a strictly decreasing function of $n$;
$(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$;
$(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$.
Prove or disprove that these conditions imply that
$$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$
a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$.
b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$.
c) Is it true that $\sqrt{2}<1.41421356$?
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.
Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
Evaluate
$$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$
For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\lim_{n\to\infty} a_n$.
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 $\langle x_n\rangle$ be a sequence of real numbers defined as
\[x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i\]
Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$