Found problems: 837
A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists.
Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.
Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit
$$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$
Let $\{x_n\}$ be a real sequence defined by:
\[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\]
For all $n=1,2,3...$ and a is a real number.
Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$.
(b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.
Let $ (a_n)_{n \in \mathbb{N}^*}$ be a sequence of real positive numbers such that $ a_n>a_0,n\in \mathbb{N}$. Prove that $ \displaystyle\lim_{n\to\infty}\displaystyle\sum_{k\equal{}0}^{n}(\frac{a_k}{a_{n\minus{}k}})^k\equal{}\infty$.
Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence?
(A)$|A\cup B|=|A|+|B|$
(B)$|A\cap B|=|A|+|B|$
(C)$|A\cup B|=|A|\cdot |B|$
(D)$|A\cap B|=|A|\cdot |B|$
Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$
It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$.
(a) Find
$$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$
(b) For which $\beta$ $(1<\beta<2)$ does the following limit exist?
$$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$
Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.
Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$
Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$
(1) Find $f(0).$
(2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$
(3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$
(4) Find $\lim_{x\rightarrow +\infty} g(x)$
Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.
Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that
$a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$
$b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$
Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval?
[i]M. Laczkovich[/i]
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$.
Then the value of $G$ is?
On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.
Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $
Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.
Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$,
$(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$.
Find $\lim_{n\to \infty}(x_n,y_n)$.
Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.
Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.
Find all functions $f: (1,\infty)\text{to R}$ satisfying
$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.
[hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution.
I have also solved by using this method.By finding $f(x^5)$ in two ways
I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]
Let $a>0$. Show that
$$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$
lies between $e^a$ and $e^{a+1}.$