Tagged Questions
3
votes
1answer
67 views
Prove that the polynomial $P(x_1,x_2…,x_n)=0$ given a set of conditions.
Let $P(x_1,...,x_n)\in\mathbb{R}[x_1,...,x_n]$ (i.e. $P$ is a polynomial of real coefficients in $x_1,..,x_n$).
We are given that $\left(\frac{\partial^2}{\partial ...
2
votes
0answers
53 views
Pi identity with sum and product
Please prove this identity
$$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
2
votes
1answer
46 views
$\lim_{x\to\infty} x^{1+1/x}-x-\log x$ and $\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$
Evaluate
$$\lim_{x\to\infty} x^{1+1/x}-x-\log x$$
and
$$\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$$
Would knowing one necessarily give the other?
1
vote
1answer
28 views
Prove that $\lim_{t\to0}(\log t)(1-(2t)^{t/2})=0$
Please prove that $(\log t)(1-(2t)^{t/2})$ tends to 0 as t tends to 0.
http://www.wolframalpha.com/input/?i=lim+t-%3E0+%281-%282t%29^%28t%2F2%29%29logt
It seems the limit converges to 0 pretty ...
6
votes
2answers
108 views
If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $[a,b]\subset f([c,d])$, how to prove there is some $[r,s]$ such that $f([r,s])=[a,b]$?
Let $f:\mathbb R\to\mathbb R$ satisfy the following:
$f$ is continuous
there exist closed intervals $[a,b]$ and $[c,d]$ such that $[a,b]\subset f([c,d])$
How to prove that there ...
3
votes
3answers
58 views
If $f:[a,b]\to[a,b]$ is increasing, continuous, and $f(a)=a$, how to prove $f(E)=E$ where $E=\{x:a\le x\le b,f(x)\ge x\}$?
Let $f:[a,b]\to[a,b]$ satisfy:
$f$ is increasing
$f$ is continuous
$f(a)=a$
If $E=\{x:a\le x\le b,f(x)\ge x\}$, then how can we prove that $f(E)=E$?
10
votes
4answers
323 views
Proving the inequality $\arctan\frac{\pi}{2}\ge1$
Do you see any nice way to prove that
$$\arctan\frac{\pi}{2}\ge1 ?$$
Thanks!
Sis.
9
votes
3answers
149 views
Proving that $x(1-e^{-1/x})$ is strictly increasing
Prove that the function below is strictly increasing
$$f(x)=x(1-e^{-1/x}), \quad x>0$$
4
votes
3answers
114 views
Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$
Find
$$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
13
votes
4answers
438 views
Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
For every nonnegative integer $n$ and every real number $ x$ prove the inequality:
$$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
4
votes
2answers
146 views
Evaluation of $\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$
Evaluation of
$$\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$$
Sis.
7
votes
1answer
117 views
$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$
let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
8
votes
1answer
143 views
Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$
Prove that
$$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$
EDIT: inspired by Michael Hardy's suggestion I got that
$$\arcsin ...
3
votes
1answer
38 views
$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$
Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
0
votes
2answers
62 views
Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.
Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U.
Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C.
Thoughts: Let ...
2
votes
5answers
126 views
Another limit from a math contest $\lim_{n\to\infty}\frac{x_n^2y_n}{3x_n^2-2x_ny_n+y_n^2}$
Let $(x_{n})_{n\ge1}$, $(y_{n})_{n\ge1}$ be real number sequences and both converge to $0$.
Evaluate $$\lim_{n\to\infty}\frac{x_n^2y_n}{3x_n^2-2x_ny_n+y_n^2}$$
4
votes
3answers
167 views
Evaluate $\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\displaystyle\frac{f(x)}{x}}$
Let's consider the function $f:\mathbb{R}\rightarrow(0,\infty)$, with $f(x)\cdot \ln f(x)=e^x$, $\forall x \in \mathbb{R}$. Then compute
$$\lim_{x\to\infty}\left(1+\frac{\ln ...
2
votes
2answers
107 views
$\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$
Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
0
votes
1answer
86 views
Convergence of series with modified denominator
Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$.
Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges,
and prove or ...
11
votes
2answers
86 views
Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
3
votes
1answer
91 views
For any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
Find the minimum possible value of $A$ such that for any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
9
votes
3answers
245 views
Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$.
Find the maximum and minimum of
$$
\sum_{i=1}^{n-1}x_ix_{i+1}
$$
subject to
$$
\sum_{i=1}^nx_i^2=1
$$
for all $n\in\mathbb{N}-\{1,0\}$.
2
votes
1answer
65 views
Limit $\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+…+\exp(ia_n)}{n}=\alpha$
Show that for any sequence $a_1,a_2,...$ of real numbers, the two conditions
$\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+...+\exp(ia_n)}{n}=\alpha$
and
...
4
votes
0answers
51 views
Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
1
vote
1answer
76 views
Find all values of $c$ such that there exists a line that intersects the graph of $f(x) =x^4+9x^3+cx^2+9x+4$ in four distinct points.
Find all values of $c$ such that there exists a line that intersects the graph of $f(x) =x^4+9x^3+cx^2+9x+4$ in four distinct points.
I considered the derivative $f'(x)=4x^3+27x^2+2cx+9$. If it could ...
19
votes
1answer
530 views
Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?
Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?
Does there exist a sequence with the same property but with each term ...
1
vote
1answer
125 views
Prove that $1-x/3\le\frac{\sin x}x\le1.1-x/4, \forall x\in(0,\pi]$
Prove that
$$1- \frac{x}{3} \le \frac{\sin x}x \le 1.1 - \frac{x}{4}, \quad \forall x\in(0,\pi].$$
60
votes
6answers
1k views
Contest problem about convergent series
The following is probably a math contest problem. I have been unable to locate the original source.
Suppose that $\{a_i\}$ is a set of positive real numbers and the series $$\sum_{n = 1}^\infty ...
3
votes
1answer
127 views
An infinite series of a product of three logarithms
I was told this very interesting question today, and despite my efforts I did not manage to get very far.
Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log ...
16
votes
2answers
1k views
Olympiad calculus problem
This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows.
Given a continuous function $f : [0,1] \to ...
