Found problems: 837
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 $\lim_{n\to\infty} c_n.$
Define a sequence $a_n$ satisfying :
\[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\]
Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$
Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]
Show that the sequence
$$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$
converges and evaluate the limit.
Define a sequence by $a_1=1,a_2=\frac12$, and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$.
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$
Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$
Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]
Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$.
a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$
b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$
Let $f:[0,1]\to\mathbb{R}$ be a continuous function and let $\{a_n\}_n$, $\{b_n\}_n$ be sequences of reals such that
\[ \lim_{n\to\infty} \int^1_0 | f(x) - a_nx - b_n | dx = 0 . \]
Prove that:
a) The sequences $\{a_n\}_n$, $\{b_n\}_n$ are convergent;
b) The function $f$ is linear.
Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.
Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.
A player repeatedly throwing a die is to play until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$
Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
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 $\lim_{n\to\infty} S_n$.
In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$
Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$
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 $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define :
$P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$
Compute $\lim_{n\to \infty} P_n$
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits.
[img]http://i2.tinypic.com/5x73dza.png[/img]
Find the limit
$$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$
Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\]
[i]P. Erdos[/i]