If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$  Since $(a_n)$ is decreasing and bounded below, by Monotonic Convergence Theorem, $(a_n)$ converges. So, there exists $N$ such that $|a_n-L|&lt; \epsilon$   Since $\sum{a_n}$ is convergent, by the Divergence test, we have $\lim_n{a_n}=0$, which means there exists $N$ such that $|a_n-0|&lt; \epsilon$  Then I get stuck at here.  I try to figure out the statement's meaning by inserting some examples, like $a_n=\dfrac{1}{n^2}$  Can anyone guide me on this question?
I'm studying for qualifying exams and ran into this problem.  Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} n a_n = 0$.  Using the definition of the limit, this is equivalent to showing \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } |n a_n| &lt; \varepsilon \; \forall n &gt; n_0 \end{equation} or \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } a_n &lt; \frac{\varepsilon}{n} \; \forall n &gt; n_0 \end{equation} Basically, the terms must be bounded by the harmonic series. Thanks, I'm really stuck on this seemingly simple problem!
Test if there are files matching a pattern in order to execute a script I am trying to write an if statement to test whether there are any files matching a certain pattern. If there is a text file in a directory it should run a given script.  My code currently:  if [ -f /*.txt ]; then ./script fi   Please give some ideas; I only want to run the script if there is a .txt in the directory. Basically, I've designed a booklet on Photoshop CC. It's an informational booklet so it's quite vibrant with lots of colours and pictures in it.   I've finished it and exported all the files into PDF format. Upon doing this, the PDF's are much "duller" for use of a better word. Now I'm not a designer and the whole thing was a learning process but now I'm reading about CMYK and RGB colour models and conversions etc. I designed the whole thing RGB (because CMYK looked duller obviously. The PDF's aren't disastrous, red is still red, black is still black but it's nowhere near as visually appealing anymore.   Is this just because of the monitor and the models each program uses or when I send my brochure to be printed, am I doomed for a much duller booklet? There's obviously a way around it otherwise all print would be dull. My new 1,5 TB ext4 Hard Disk makes mysterious noises and fails to read and write. But not always. Sometimes it just works. But then I get those following errors. Does anyone know a solution for that or is my new WD Caviar Black 1.5 TB hard disk broken?  [  551.162286] ata7: SATA link down (SStatus 0 SControl 310) [  551.162294] ata7.00: link offline, clearing class 1 to NONE [  551.178223] ata7: EH complete [  551.190802] ata7: exception Emask 0x10 SAct 0x0 SErr 0x4040000 action 0xe frozen [  551.190807] ata7: irq_stat 0x00000040, connection status changed [  551.190811] ata7: SError: { CommWake DevExch } [  551.190818] ata7: limiting SATA link speed to 1.5 Gbps [  551.190822] ata7: hard resetting link   Update Mainboard P7P55D LE: Not sure if there is a cultural difference. In our family it is considered normal to frighten the child so that he behaves the way we want.  Example- Parent: Eat that quickly or the God will punish you! Parent: Don't go on the stairs. The ghost is hiding there and it'll catch you! Parent: Be quiet or the police will grab you!  Does this kind of behavior from the parent's side has any abnormal effects on the baby in the longer run? Can this behavior be accepted as normal?  Has any research been done on this aspect? Double decimal formatting in Java Aligning equations within multline I have this really long equation and I have used multline to break it up. I also want to align the broken bits of the equation to the left. When I use  align, for some reasons it aligns to the right when I want it to be on the left. How can I align to the left?  \begin{multline} \begin{align} \mathcal{H}_{q1} \mathcal{H}_{q2} \mathcal{H}_{q3}...\mathcal{H}_{qn}0=\frac{1}{2}H_{q1}\{\mathcal{H}_{q2}\mathcal{H}_{q3}...\mathcal{H}_{qn}0\}+\frac{1}{2} h_{q1} \\ =\frac{1}{2}H_{q1}\{\frac{1}{2} H_{q2}[\mathcal{H}_{q3}...\mathcal{H}_{qn} 0]+\frac{1}{2}h_{q2}\} +\frac{1}{2}h_{q1}=\\ \frac{1}{2}H_{q1} \frac{1}{2} H_{q2} [\frac{1}{2}H_{q3}(\mathcal{H}_{q4}..\mathcal{H}_{qn}0)+\frac{1}{2}h_{q3}]+ \frac{1}{2}h_{q2}+\frac{1}{2}h_{q1}=...\\          =(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn}0+ (\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-1}h_{q1}\\+(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-2}h_{qn-1}+.. \\+ (\frac{1}{2})h_{q1}=\sum^n_{j=1} (\frac{1}{2^j}) H_{qo}H_{q1} H_{q2}H_{q3}..H_{qj-1} h_{qj} \end{align} \end{multline} customize of autoref Abnormal PREEMPTIVE_OS_AUTHENTICATIONOPS Waits How to move my cube left or right using touch Inputs (C#) in unity I am trying to create a 3d game in unity. I have a cube in unity and I want it to move left or right on swipe . It should move to a distance till where user swipe or take their finger off of a screen. I want it to move fast or the cube should move fast on swipe and cube should flow smooth. Like in the gif there is a circle and the user use their finger to move the circle till wherever the user swipe it and it should be instant and I want my cube to move like that at a distance till where user swipe their finger and it should be instant and fast.  I don't know how to do it. Any help is appreciated! With the last update of Fallout Shelter, Bethesda has added these "limited time" quests.  Question is, how limited is that time exactly? What happens when the quests "end", will it still be possible to get the benefits? I am unable to print documents using my HP printer. However, I am able to print test pages.  What I tried:  I have tried removing and re-installing the printer from the system settings but I can only print once (the first task). Then all the following tasks get stuck at the "Rendering Completed" status.  I additionally tried restarting the computer which leads in the same task mentioned above.  Moreover, I logged into the CUPS system ( 1.6.2) from http://localhost:631  From there, I removed the printer, restarted it and re-installed with no sucess.  System Specifications:  System : Ubuntu GNOME 13.04  Memory: 3.9 GB  CPU : AMD Athlon(tm) 64 X2 Dual Core Processor 4200+ × 2   OS-type : 32 bit  Graphics: Gallium 0.4 on NVA3  Printer: HP Deskjet 2050 j510 Series  IP Adress: localhost  I had no problems with my previous OS which was Pinguy OS 11.04 ( a distro based on Ubuntu 11.04) and it's the first time I use my printer at this OS.  Screenshots:   My printer stuck at "Rendering Completed" status.     From the CUPS web interface; you can see that "Rendering Completed" is stuck at the state of the print process.  **EDIT:**I tried printing from the pdf.js pdf viewer of firefox: I wait for about 5 minutes for a 10 pages document to print, the document is printed like faded and sort of in bold font (I accuse rendering for that). Then the status of the printer gets stuck at 'Rendering Completed' but I can print later on. Again, for the next prints I wait for 5 minutes minimum ( I wait for the 'Rendering....' bar to fill ) and the printing is weird. Counting binary operations on a set with $n$ elements I am trying to solve following problem but not able to find any way to proceed.  Let $S$ be a set having $n$ elements. Can we count about number of binary operations that can be defined on a set? Can we also count number of commutative binary operations defined on  $S$?  Thanks for the help and suggestions Why is the copy-constructor argument const? What is this shell/Bash syntax: someVariable=someValue someCommand Inspired by this Hagon Von Eitzen's answer( ) I started investigating how I could express natural numbers as differences of squares. Using the method explained in the solution, I investigated natural numbers up to $60$ manually. I observed the following:  There is no way to express numbers of the form $4k+2$ as a difference of two squares.  2.There is one and only one way to express prime numbers (except $2$) as a difference of two squares.  The smallest number that can be expressed as a difference of two squares in two different ways is $15$ and the smallest number that can be expressed as a difference of two squares in three different ways is $45$.  As, $15= 8^2-7^2=4^2-1^2$. $45=23^2-22^2=9^2-6^2=7^2-2^2$. I could prove point $2$ easily but could not prove point $1$. Also I am interested in knowing how I could find the smallest natural number that could be expressed as a difference of two squares in four different ways without manual calculation. Prove that no ordinal is an element of itself Is it possible to convert a Physical Ubuntu Install (Intel based PC with Ubuntu 10.10) to a virtual box disk, so that I can use that same environment on a separate pc or can keep it virtual for testing?  Something similar to "Disk2VHD" which converts a physical Windows 2K3 and above box into a VHD or VMware image.  Will be grateful for your assistance ! Kind Regards If I have an i.i.d. random variables $X_i$ from a uniform distribution on $[0,1]$.   How would I find scaling sequences $a_n$, $b_n$ such that $a_n(M_n − b_n)$ converges in distribution to a non-trivial limit function $G$ for  $ Y_i = X_i,$ and $M_n = \max(Y_1,...,Y_n) $ In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). Sex worker student offering her (lawful) services One of my students is an escort/stripper and she has offered me (and other faculty in my department) her "services". I am pretty sure my initial reaction of "thanks for letting me know, now can you answer the question about how to calculate the standard deviation", may not have been the best reaction. The offers have continued. My head of school is aware of the issue, and has asked if I want him to do anything.  Is the student doing anything wrong by offering her services to me? For what it is worth, prostitution is legal in the UK. Is this any different then a student telling me she works in a restaurant?  What is the correct response in a situation like this?