Let $(x_{n})_{n}$ be a sequence of positive numbers for which the corresponding series is convergent. How can I show that there exists a sequence $(y_{n})_{n}$ converging to infinity such that the corresponding series of $(x_{n}y_{n})_{n}$ is convergent?  I am only interested in the existence of such sequence, but an explicit example is welcome :)
This is a nice problem on series convergence that I recently stumbled upon. Given a non-negative sequence of real numbers $(a_n)$ such that $$\sum_{n=1}^\infty a_n &lt; \infty,$$ show that there exists a non-decreasing sequence of non-negative numbers $b_n$ such that $$b_n \to \infty \quad\text{ and } \quad \sum_{n=1}^\infty a_n b_n &lt; \infty.$$ In other words, for every convergent series with non-negative terms, there is another convergent series with "substantially larger" terms.  I have a solution (see below), but maybe someone else has a different, simpler, and/or more elegant solution.
The merits of a partitionless filesystem I ran into something a couple of weeks ago that I'd never seen before: A filesystem (ext3 I believe) installed to a storage device without a partition. In essence /dev/sdb was the entire filesystem. I know many filesystems can be extended into empty space, so doing this allows extending without dealing with LVM or some other kind of volume manager, but are there any other advantages for setting up storage this way?  The specific case I saw was as the ephemeral data volume for a number crunching server, the boot and root volumes were traditional partitions on a different storage device entirely. - Changing Unsubscribe reason through SSJS I noticed that you can unsubscribe a user , is it possible to use SSJS to change the unsubscribe reason?  I notice that doing it through ? why is that?  Thanks I have an empty file (only zeroes are in it) of size 9,0KB and I need to write another file (with size 1,1KB) to it, but the first file must not lose its size or the rest of its contents. So if the whole file is 00000000000000... now, I need to write second file in it and leave the zeroes as they are. I have tried to use dd, but I haven't succeed - file resizes.  dd if=out/one.img of=out/go.img   Does anybody know how can I do it? $\operatorname{Aut}(V)$ is isomorphic to $S_3$ I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks... What is the difference between "gender" and "sex"? What is the difference between gender and sex?  says that gender is The mental analog of sex but that's too high English for me.  Basically, I'm developing a web-application that stores people's profile along with a field for gender/sex. Which name would be more appropriate?  __  [UPDATE]  After getting these good answers, I can figure out that,   Sex - Biological characteristics  Gender - Social characteristics   And, well, for my web-application, I went with the term sex, as it sounds more scientific and less political. P.S. Values for sex are clearly defined and can be easily validated (only two values * + an optional blank value).  * plus some anomalies. Can we create a subsection of the References Cited section with different reference numbering? Solving "mv: Argument list too long"? Shell script- To fetch the content from the specific tag at all occurrence using shell scripting I have an XML file on my server. In that file, I have many tags in which there is a tag for example, "container" tag. I need to fetch the content of all the container tags content separately. Many thanks in advance. Using CLI, how do I change the user's profile picture? A similar task for changing the desktop background, can be achieved using the following script:   FILE="/backgrounds/background.png" gsettings set org.gnome.desktop.background picture-uri "$FILE"   How would I achieve the same result, but with the user's avatar/profile picture? Every $R$-module is a quotient of a free module I'm reading through my notes, and I have a theorem as the title says. As proof I have only a picture.    where $v:m\mapsto(0,...,0,1,0,...,0)$ and 1 is at $m$-th position. I know, that $\oplus_M R$ is a free module generated by set  $M$, but is this all? I do not see how the statement is proved. Why copy constructor not invoked here I try to clean my mid-2015 Macbook Pro with Retina display by wiping the screen with a damp cloth and then immediately drying.  However, there seems to be left some residue/streak marks, which look very oily. Sometimes these disappear after a few minutes, but they are still very unsightly, and a bit worrying (is there something wrong with my screen?)  I'm wondering if there are other cleaning techniques which avoid this issue, or whether this is completely normal behaviour.  Thank you! How can I make environment variables "exported" in a shell script stick around? At WWDC 2018 macOS 10.14 or macOS Mojave was disclosed.  I am currently on macOS Sierra (10.12) and am looking forward to upgrading to macOS Mojave once its released.  I was wondering if it will be possible to update to macOS Mojave without converting to APFS, similar to  trick from OS X Daily on how to upgrade to macOS High Sierra (10.13) without converting to APFS with the command /Applications/Install\ macOS\ High\ Sierra.app/Contents/Resources/startosinstall --converttoapfs NO. Imagine that you have the following list. keys = ['name', 'age', 'food'] values = ['Monty', 42, 'spam']  What is the simplest way to produce the following dictionary? a_dict = {'name': 'Monty', 'age': 42, 'food': 'spam'} It maby be weird, I want to make header and footer for every page including the first page of each chapter, how can I do this? Syntax to add multiple commands to a single command block I am looking to try and set multiple commands in a single command block, but so far everything I have tried just returns "Unknown Command".  What is the format for writing multiple commands in a single command block in Minecraft 1.8?  Can I use either of the following:  Command:(command here) Command:(command here)     /command here; /command here;   Or is there another format? Force Messages to use SMS for specific contact rather than iMessage? Essentially, I would like to set my phone so that all my outgoing messages to a specific friend, with whom I currently text back and forth over iMessage, are sent out over SMS, never iMessage. Additionally, I would like to block any incoming iMessage texts of his and (hopefully) they'll fail and be resent via SMS.  Let me restate that: I would like to force SMS for both parties, going both ways, for a specific individual contact.   Is this possible? Blocking any usage of iMessage between myself and this other individual, so they must either send and receive any texts to or from me via SMS (or the texts fail/never arrive). Passing directly an array initializer to a method parameter doesn't work What do the numbers in a man page mean?