Search found 179 matches
- Wed Nov 29, 2017 9:20 am
- Forum: Algebraic Structures
- Topic: True or false statements
- Replies: 1
- Views: 4920
True or false statements
Let $n \in \mathbb{Z}$ such that $n \geq 2$. Let $\mathcal{S}_n$ be the permutation group on $n$ letters and $\mathcal{A}_n$ be the alternating group. We also denote $\mathbb{C}^*$ the group of non zero complex numbers under multiplication. Which of the following are correct statements? For every in...
- Thu Nov 02, 2017 9:35 pm
- Forum: Real Analysis
- Topic: On the evaluation of the Fresnel integral
- Replies: 0
- Views: 3188
On the evaluation of the Fresnel integral
We are aware of the Fresnel integral \begin{equation} \int_0^\infty \sin x^2 \, {\rm d}x = \frac{1}{2} \sqrt{\frac{\pi}{2}} \end{equation} The most common proof goes with complex analysis. Try to provide a proof with Real Analysis. There are at least $2$ proofs. The one is more elegant than the othe...
- Fri Oct 13, 2017 9:35 pm
- Forum: Number theory
- Topic: Irreducible number
- Replies: 0
- Views: 4870
Irreducible number
Denote by $\bar{\alpha}=0.\alpha \alpha \alpha \dots $. Find all $\alpha>0$ such that
$$\frac{1}{\alpha} = 0.\bar{a}$$
$$\frac{1}{\alpha} = 0.\bar{a}$$
- Wed Sep 13, 2017 8:46 am
- Forum: Calculus
- Topic: On a prime summation
- Replies: 0
- Views: 3358
On a prime summation
Let $p_n$ denote the $n$ -th prime number. Evaluate the sum
$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{ \log p_n}{p_n^2 -1}$$
$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{ \log p_n}{p_n^2 -1}$$
- Sat Sep 09, 2017 8:17 am
- Forum: Real Analysis
- Topic: Limit of an integral
- Replies: 1
- Views: 2810
Re: Limit of an integral
Since $\lim \limits_{x \rightarrow +\infty} f(x) =0$ the result follows immediately by making the change of variables $u=nx$ .
- Sat Sep 09, 2017 8:12 am
- Forum: Meta
- Topic: MathJaX Upgrade
- Replies: 1
- Views: 6077
Re: MathJaX Upgrade
\begin{xy} <0.3pc,0pc>:(0,0) *+{0} \PATH ~={**@{-}} ~<{|<*@{<}} ~>{|>*@{>}} '(10,1)*+{1}^{a} '(20,-2)*+{2}^{b} (30,0)*+{3}^{c} \end{xy} \begin{xy} <4pc,0pc>:(0,0) *+{base}="base" \PATH ~={**@{-} ?>*@{>}} `l (-1,-1)*{A} ^a ` (1,-1) *{B} ^b `_ul (1,0) *{C} ^c `ul^l "base" ^d "...
- Tue Aug 22, 2017 9:07 am
- Forum: Real Analysis
- Topic: On an inequality of a product function
- Replies: 0
- Views: 2671
On an inequality of a product function
Let
$$f(x) = \sin x \sin (2x) \sin (4x) \cdots \sin (2^n x)$$
Prove that
$$\left| {f(x)} \right| \le \frac{2}{{\sqrt 3 }}\left| {f(\frac{\pi }{3})} \right|$$
$$f(x) = \sin x \sin (2x) \sin (4x) \cdots \sin (2^n x)$$
Prove that
$$\left| {f(x)} \right| \le \frac{2}{{\sqrt 3 }}\left| {f(\frac{\pi }{3})} \right|$$
- Sun Aug 20, 2017 9:25 am
- Forum: General Mathematics
- Topic: Inequality
- Replies: 2
- Views: 6082
Re: Inequality
Thank you Papapetros Vaggelis. My solution is as follows. Since $\frac{1}{x} \; , \; \frac{1}{y} \; , \; \frac{1}{z} >0$ then the numbers \[\sqrt{\frac{1}{x} + \frac{1}{y}} \; , \; \sqrt{\frac{1}{x} +\frac{1}{z}} \; , \; \sqrt{\frac{1}{y} + \frac{1}{z}}\] could be sides of a triangle. The area of th...
- Sun Aug 13, 2017 4:55 pm
- Forum: General Mathematics
- Topic: Inequality
- Replies: 2
- Views: 6082
Inequality
Let $x, y,z >0$ satisfying $x+y+z=1$. Prove that
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \sqrt{\frac{3}{xyz}}\]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \sqrt{\frac{3}{xyz}}\]
- Sat Aug 12, 2017 4:29 pm
- Forum: Calculus
- Topic: Dobiński’s formula
- Replies: 0
- Views: 3298
Dobiński’s formula
Let $n \in \mathbb{N}$ and $\mathcal{B}_n$ denote the $n$ - th Bell number. Prove that
$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$
$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$