My question is as follows: Does there exist a sequence $(a_n)$ of non-negative real numbers such that $\sum_{n=1}^\infty a_n &lt; \infty$, but for any sequence $(b_n)$ of non-negative numbers with $b_n \to \infty$, we have $\sum_{n=1}^\infty a_nb_n = \infty$? To me this seems to be asking if there is in some sense a &quot;maximal&quot; convergent series, as any &quot;bigger&quot; series diverges, where &quot;bigger&quot; means pointwise multiplied by some unbounded non-negative sequence. Thus my first thought to solve this was Zorn's lemma. I tried to define an order $\preceq$ on the space of all convergent non-negative sequences by saying $(x_n) \preceq (y_n)$ if there exists an unbounded sequence $(b_n)$ of non-negative numbers such that $y_n = b_nx_n$ for all $n$. But this order is not reflexive so we cannot use Zorn's lemma. I did manage to prove a weaker version, namely that if $(b_n)$ is any unbounded non-negative sequence then we can find a non-negative sequence $(a_n)$ with $\sum_{n=1}^\infty a_n &lt; \infty$ and $\sum_{n=1}^\infty a_nb_n = \infty$, and I can easily extend it to finitely many $(b_n)$, but have not been able to extend it to infinitely many (i.e. for all unbounded sequences). Any suggestions are appreciated!
Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges.  Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ converges.  What I have done:  I have written $$\sum_{i=1}^{N+1}a_ic_i=c_{N+1}\sum_{i=1}^{N+1}a_i-\sum_{i=1}^N(c_{n+1}-c_n)S_n$$  and then try to prove the convergence by the Cauchy criterion. However my evaluation is still not good enough. I'm really stuck.
Taking the second derivative of a parametric curve Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. It is my understanding that I can disable my touchscreen by adding Option "Ignore" "on" to this file                                /usr/share/X11/xorg.conf.d/10-evdev.conf   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   so it would look like this   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           Option "Ignore" "on"           EndSection   What would happen if I simplied changed MatchIsTouchscreen to "off"?   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "off"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   Would this disable my touchscreen? “situation where” vs. “situation in which” It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. How does the Deposit-User work? On SU there's the  which is suspended for quite some time. And I also know that it is used for spammers, but how does it work? Does the account of the spammer get 'linked' to Deposit? Do the answers get moved there? In Harry Potter, we know ghosts became ghosts when they are too afraid of death to go on - I thought about it and I'm quite certain they all go to their own version of Harry's King's Cross and become ghosts when they return as Harry did.  So can they return there? Once they've been a ghost long enough that they no longer fear death, can they move on? I've found plenty of posts about changing the visibility of a block programatically.  My problem is the opposite. Given a block's ID/delta, how do I find out if it should be displayed at the current path?  I have a field collection that allows you to append a block using the Block Reference module (I preprocess the field collection and use my own template, so the logic has been taken away from the Block Reference module). It works, but I only want it to kick in if the "Show block on specific pages" rule says that block should be displayed. What is the name of a squircle with three, five, six or more corners? I know that the second shape is a squircle, but what are the other shapes called? Is there an actual name for them?    The shapes do not simply have rounded corners, but the sides have a "circularity". Here is a triangle with rounded corners next to the shape that I do not know the name of:    "Trircle", "Triarcle", and "Pentircle" do not seem to be used for them, at least Google's image search does not provide any results. When & How often do 4e Monster powers recharge? How often should I roll power recharge rolls and at which point in the monster's turn do I re-roll them?  I'm trying to find the official definition of the recharge mechanic in the online compendium and I'm not seeing it there. I have a batch file which I use to create an .apk file on my Windows machine. Now I need to be able to create the .apk file in Ubuntu but I don't know how translate my .bat file to a script to be able to run it on Ubuntu.  Below is the batch file which works fine on Windows.  Will you please give me some hints on how I can run it on Ubuntu?  @echo off set PAUSE_ERRORS=0 :user_configuration  :: Path to Flex SDK  set FLEX_SDK=C:\sdk\flex_sdk_4.5.1.21328  :: Path to Android SDK set ANDROID_SDK=C:\sdk\android  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :flexsdk echo. echo ERROR: incorrect path to Flex SDK echo. if %PAUSE_ERRORS%==1 pause exit  :androidsdk echo. echo ERROR: incorrect path to Android SDK in 'bat\SetupSDK.bat' echo.  if %PAUSE_ERRORS%==1 pause exit  :succeed set PATH=%PATH%;%FLEX_SDK%\bin set PATH=%PATH%;%ANDROID_SDK%\platform-tools  :: Android packaging set AND_CERT_NAME="PeymanApp" set AND_CERT_PASS=fd set AND_CERT_FILE=cert\SampleApp.p12 set AND_ICONS=icons/android  set AND_SIGNING_OPTIONS=-storetype pkcs12 -keystore "%AND_CERT_FILE%" -storepass     %AND_CERT_PASS%  :: Application descriptor set APP_XML=application.xml  :: Files to package set APP_DIR=bin set FILE_OR_DIR=-C %APP_DIR% .  :: Your application ID (must match &lt;id&gt; of Application descriptor) set APP_ID=air.com.doitflash.SampleApp  :: Output packages set DIST_PATH=dist set DIST_NAME=PeymanApp  :validation %SystemRoot%\System32\find /C "&lt;id&gt;%APP_ID%&lt;/id&gt;" "%APP_XML%" &gt; NUL if errorlevel 1 goto badid goto end_validation  :badid echo. echo ERROR: Application ID (APP_ID) does NOT match Application descriptor '%APP_XML%'     (id)  echo.  :end_validation  set TARGET= set PLATFORM=android  ::call bat\Packager.bat if "%PLATFORM%"=="android" goto android-config  :android-config set CERT_FILE=%AND_CERT_FILE% set SIGNING_OPTIONS=%AND_SIGNING_OPTIONS% set ICONS=%AND_ICONS% set DIST_EXT=apk set TYPE=apk goto start  :start if not exist "%CERT_FILE%" goto certificate :: Output file set FILE_OR_DIR=%FILE_OR_DIR% -C "%ICONS%" . if not exist "%DIST_PATH%" md "%DIST_PATH%" set OUTPUT=%DIST_PATH%\%DIST_NAME%%TARGET%.%DIST_EXT% :: Package  echo true echo. call adt -package -target %TYPE%%TARGET% %OPTIONS% %SIGNING_OPTIONS% "%OUTPUT%"     "%APP_XML%" %FILE_OR_DIR% -extdir lib/ echo. if errorlevel 1 goto failed goto end  :certificate echo Certificate not found: %CERT_FILE% echo.  if %PAUSE_ERRORS%==1 pause exit  :failed echo APK setup creation FAILED.  if %PAUSE_ERRORS%==1 pause exit  :end How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? Terms for collections of animals As I watched the murder of crows sitting on the line above my house this evening, I got wondering where all of the collective nouns for animals (pod of whales, gaggle of geese, pride of lions) came from and why we need so many.    If sheep can be a flock, why can't whales, geese, lions, and crows? How to make a cylinder / pipe with slanted ends (like pasta penne)? Do systemd unit files have to be reloaded when modified? Let's say I write a mine.service file. Then I use systemctl enable mine.service.  If I later decide to edit mine.service, do I have to tell systemd that mine.service was changed? If so, how do I do that? How does an attacker's reach interact with the Mounted Combatant feat's ability to force an attack against the mount to target the rider instead? How to open the default webbrowser using java This is a  about Apache's mod_rewrite.   Changing a request URL or redirecting users to a different URL than the one they originally requested is done using mod_rewrite. This includes such things as:   Changing HTTP to HTTPS (or the other way around) Changing a request to a page which no longer exist to a new replacement. Modifying a URL format (such as ?id=3433 to /id/3433 ) Presenting a different page based on the browser, based on the referrer, based on anything possible under the moon and sun. Anything you want to mess around with URL   Everything You Ever Wanted to Know about Mod_Rewrite Rules but Were Afraid to Ask!  How can I become an expert at writing mod_rewrite rules?   What is the fundamental format and structure of mod_rewrite rules? What form/flavor of regular expressions do I need to have a solid grasp of? What are the most common mistakes/pitfalls when writing rewrite rules? What is a good method for testing and verifying mod_rewrite rules? Are there SEO or performance implications of mod_rewrite rules I should be aware of? Are there common situations where mod_rewrite might seem like the right tool for the job but isn't? What are some common examples?   A place to test your rules  The  web site is a great place to play around with your rules and test them. It even shows the debug output so you can see what matched and what did not. Logistic regression is predicting all 1, and no 0 I am running an analysis on the probability of loan default using logistic regression and random forests.    When I use logistic regression, the prediction is always all '1' (which means good loan).  I have never seen this before, and do not know where to start in terms of trying to sort out the issue.  There are 22 columns with 600K rows.  When I decrease the # of columns I get the same result with logistic regression.   Why could the logistic regression be so wrong?  **Actual from the data**  0 :   41932  1 :   573426  **Logistic regression output**   prediction for 1 when actually 0: 41932 prediction for 1 when actually 1:573426  A**s you can see, it always predicts a 1**   **Random forests does better:**  actual 0, pred 0 : 38800  actual 1, pred 0 : 27  actual 0, pred 1 : 3132 actual 1, pred 1 : 573399 How can I give a script its own icon in the Unity launcher? With Ubuntu 16.04, if you create a launcher (desktop file in ~/.local/share/applications) with a custom icon for a terminal application (so with Terminal=true) then when you start it, it will spawn a new standard terminal icon, and your custom icon will pulse and disappear within a few seconds.  In 14.04, it just worked as expected. (it would not start a new standard terminal icon).   Any idea what to do to change this behaviour? I have a few terminal apps that I want to launch from unity and the new behaviour is problematic (I lose trace of which is which since they all end up with the standard terminal icon)...