Found problems: 837
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$
where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate
$\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$
Evaluate
\[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]
Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$
If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by:
\[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \]
Show that there exist a real number $a > 0$ such that:
[b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$
[b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.
For any positive integer $k$ consider the sequence
$$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$
where there are $n$ square-root signs on the right-hand side.
(a) Show that the sequence converges, for every fixed integer $k\ge 1$.
(b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$.
(1) Find $ I_1,\ I_2$.
(2) Find $ \lim_{n\to\infty} I_n$.
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by
\[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\]
and $x_0,y_0$ are given real numbers.
a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit.
b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.
Find the limit as below.
\[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$
Find the following limit.
$ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
Let $n\ge 2$ be a positive integer. Find the positive integers $x$
\[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \]
for any number of radicals.
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$.
Answer the following questions.
(1) Find $ a_1$.
(2) Express $ a_{n\plus{}1}$ in terms of $ a_n$.
(3) Find $ \lim_{n\to\infty} a_n$.
(4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.
Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$.
[list=1]
[*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$
[*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$
[/list]
Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.
If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$
Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$
Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$
A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$:
\[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\]
Describe, with proof, the behavior of $u_n$ as $n \to \infty.$
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.