Am I able to prove the monotonicity with induction?
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.   I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$  I am trying to go ahead of this step.
Currently, for all my commands that require root access, I have to type them out on a command line.  I would like for the GUI to prompt me to key in my password, when sudo is required, instead of having to type out the commands from command line.  Is there any way to enable this? Previously one version of Ubuntu desktop had this feature out of the box.  This version of Ubuntu (that I am now running) I upgraded from a server installation.  I will add, my user account is already added to sudo group. Thus I can run commands such as sudo gedit or sudo apt-get but I'm not able to install software in the Ubuntu Software Center. can we have a parallel earth made of dark matter? How do you parse and process HTML/XML in PHP? Deleting self-answered question gets "Delete this answered question" warning How to disable USB storage device support I want to disable a particular USB storage device (a sandisk flash drive). I read somewhere that you can use the .config file. How do i use it to disable that? Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does? Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom?  Why can the first order Dirac equation explain it? What are the failures? Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. In MySql, find strings with a given prefix Divergence of $\sum\limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$ Background info: I am planning on making a filehost with which one can encrypt and upload files. To protect the data against any form of hacking, I'd like not to know the encryption key ($K$) used for a file, so the user will have to asymmetrically encrypt $K$ and send it as well. The reason for this double encryption is that you can still share (individual!) files with other accounts by encrypting $K$ with the public keys of those users, too.  My problem is: How can a user can get a public and private key without the server saving/knowing the private key and without the user having to remember a passphrase of 250 hex characters?  Question: Is there a way to derive a key pair from a password, like PBKDF-2 does for symmetric encryption? Suggestions on rearranging the cryptographic steps in an easier way are welcome, too.  Alternatively: It is also possible to encrypt the private key with a PBKDF-applied password and store it on the server, but I'd like to keep things 'simple'. ;)  Note: At this point I don't care if it requires RSA, ECC or an obscure cryptography scheme. $ not defined using jQuery in WordPress I'm stuck with the proof of the following:  Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.   I've tried starting with $s_n \leq t_n \: \forall : n$ and the definitions of each limit (i.e. $|s_n - s| \leq \epsilon \: \forall \: n &gt; N_1$), but I'm not really getting very far. Any help is appreciated! How to manipulate a CSV file with sed or awk? How can I do the following to a CSV file using sed or awk?   Delete a column Duplicate a column Move a column   I have a big table with over 200 rows, and I'm not that familiar with sed. What are bitwise operators? I'm using classicthesis to write my document. I don't use the option drafting but I can see on the bottom of the page the date (there is a part of the date that is not readable, but when I copy and paste this part I obtain [ December 1, 2011 at 20:50 ]). This is my code:  \documentclass[a4paper]{scrreprt}   % classe report di KOMA-Script  \usepackage[english]{babel}     % Per scrivere in inglese \usepackage[T1]{fontenc}        % Imposta la codifica dei font \usepackage[utf8]{inputenc}     % Lettere accentate da tastiera \usepackage{classicthesis-preamble} % Preambolo classicthesis \usepackage{classicthesis}      % Classicthesis  \begin{document}  \input{Chapters/0_Introduction} [...]  \end{document}   I would like to avoid this little issue.  I use TeXworks in Windows7 x64. When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges? This is just a general question. I know the opposite statement is always true. I'm just asking when the converse holds. I know that the converse is not true in some cases, such as for harmonic series. What additional restrictions are needed to make this statement true? My objects are semi-transparent, and not one behind the other in the compositor Repeating something with (1/n)th chance of success n times Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every guess, giving in the end a probability of 1.  If the number changes every guess tough, you just have a 1/1000th chance everytime. Can it be calculated how many attempts it would probably take, on average?  (I'm sorry, i'm not very skilled in probability maths) Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is:  Given that $P(B) = 0$,  $$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{0}{0}$$  Is this probability vacuously $0$ of $1$? Can we show that it's one or the other? Find the value of : $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$