Found problems: 837
Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.
For $n=0,\ 1,\ 2,\ \cdots$, let
$a_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}(x-n)\}\ dx,$
$b_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}\}\ dx.$
Find $\lim_{n\to\infty} \sum_{k=0}^n (a_k-b_k).$
$S$ 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 $n \in \mathbb{N}, f(n) < M$.
Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $
Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.
For $n\in\mathbb N$, define
$$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$
(a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit.
(b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.
We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$.
(a) Prove that for every natural $n$ holds: $x_n > \frac12$
(b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
Evaluate
$$\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}$$
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 $x,y$.
Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.
Find all functions $f$ from the set of real numbers to itself satisfying
\[ f(x(1+y)) = f(x)(1 + f(y)) \]
for all real numbers $x, y$.
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.
Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\]
[i]G. Halasz[/i]
How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$
(A)$20$
(B)$15$
(C)$9$
(D)$8$
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
Calculate$$\lim \limits_{n \to \infty}\frac{n!\left(1+\frac{1}{n}\right)^{n^2+n}}{n^{n+\frac{1}{2}}}$$
For real $a > 1$ find$$\lim \limits_{n \to \infty}\sqrt[n]{\prod \limits_{k=2}^n \left(a-a^{\frac{1}{k}}\right)}$$
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions:
$ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $
$ \text{(ii)}\quad f $ has Darboux’s property
[b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist.
[b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?
$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$.
Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture.
(yes, $cd$ is parallel to $a_1b_1$ there)
Show $A_k < S$ for every positive integer $k$.
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]
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 \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]
For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points