If $a_j&gt;0$ for every $j$ and if $\sum_{j=1}^\infty a_j$ converges then prove that $\sum_{j=1}^\infty \sqrt{a_j a_{j+1}}$ converges. I want to use arithmetic mean${}&gt;{}$geometric mean, but I don't know how to show the series $\sum \frac{a_j +a_{j+1}}{2}$ converges.
Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent.  Prove that if the positive term series $\sum^{\infty}_{n=1}a_n$ and $\sum^{\infty}_{n=1}b_n$ are convergent, also $\sum^{\infty}_{n=1}a_nb_n$ is convergent.  I've tried to solve it using comparison test, but no results.
What changes when an Alienist becomes an outsider? Why didn't Voldemort murder the Dursleys? Voldemort couldn't kill Harry as long as he called the Dursleys home. Why didn't he just murder the Dursleys? This would prevent Harry from being able to live there. Mean of a Cauchy Distribution Display header and footer in all PDF pages using VF My vf page     &lt;apex:page renderAs="pdf" Controller="testpdf1"&gt;     &lt;head&gt;         &lt;style type="text/css" media="print"&gt;             @page              {                 size: 8.5in 11in;/* width height */             }              @page              {                 @top-center                  {                     content: element(header);                 }                  @bottom-left                  {                     content: element(footer);                 }             }              div.header              {                 padding: 10px;                 position: running(header);             }              div.footer              {                 display: block;                 padding: 5px;                 position: running(footer);             }              .pagenumber:before              {                 content: counter(page);             }              .pagecount:before              {                 content: counter(pages);             }         &lt;/style&gt;     &lt;/head&gt;      &lt;div class="header"&gt;             &lt;div align="right"&gt;Header&lt;/div&gt;     &lt;/div&gt;      &lt;div class="footer"&gt;         &lt;div&gt;Page &lt;span class="pagenumber"/&gt; of &lt;span class="pagecount"/&gt;&lt;/div&gt;     &lt;/div&gt;      &lt;div class="content"&gt;         &lt;p&gt;             &lt;apex:repeat value="{!opp}" var="item"&gt;                 &lt;tr&gt;                     &lt;td class="tableContent"&gt;{!item.Name}&lt;/td&gt;                 &lt;/tr&gt;             &lt;/apex:repeat&gt;         &lt;/p&gt;     &lt;/div&gt; &lt;/apex:page&gt;   Controller  public class testpdf1 {     public list&lt;Account&gt; opp{get;set;}      public testpdf1()     {         opp=[SELECT Name  from Account limit 50];     } }   I am getting all records in pdf format but able to get header in first page only and i am not getting header from second pages how can i display header to all pages. Why do I get 'HDIO_GET_IDENTITY failed' message when booting with external usb hard drive? I've installed Ubuntu 11.10 on an external USB (2.0) Hard Drive (/dev/sdb1) but when Ubuntu starts I get a message that says:     ata_id[279]: HDIO_GET_IDENTITY failed for '/dev/sdb': Invalid argument   where I made a mistake? The first HD contains Windows 7 (the official OS of my packard bell notebook). How can I get a history of running processes Post Feedback link is broken I first noticed this a while back. But it has yet to resolve itself.  When I click on the  in the 10k tools, it hangs for a while.Then it gives me an error message: How can I play around targeted discard? Many decks in competitive Magic have access to targeted discard spells, like  and .  Against counterspells, you can rely on a variety of tricks like baiting counters, bottlenecking your opponent's mana, and good old-fashioned counter wars. Targeted discard cards allow your opponent to attack your hand on the very first turn of the game, and go right for your key cards &mdash; and it can often be backbreaking when your opponent takes away a crucial card like a combo piece or a sweeper.  What tricks and tactics can you employ, either during deck-building or in play, to play around an opponent's targeted discard?  (I know that one of the answers is going to be . That's excellent, but please also include approaches that work in formats without free countermagic.) How do you find the transfer function of a controller? Below is a block diagram of the system. I thought the circle symbol with the plus and minus sign was an op-amp but I'm not sure whether that's the case. I want to find the closed loop transfer function. If there was no feedback (open loop), then I think I could find the output as Y(s) = Vin*G. This would mean that the transfer function is Y(s)/Vin = G.  Any ideas for how to find the closed loop transfer function and what the circle means? Using the residue theorem, calculate $\int_{0}^{2\pi}\frac{1}{1-2a\cos{\theta}+a^2}d\theta$ I want to prove the assertion:      The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.   My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit.  Since $p$ is an odd prime, $(p-1)/2$ is an integer and  $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.    How do I deal with the $\epsilon$ under the square root?  I guess the condition on the congruence class of $p$ comes from that.  Is this even the right way to proceed?  Uniqueness is not clear to me either.  I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.  I would appreciate some help. Maybe I'm searching for the wrong thing, but I can't see how I can do this.  I have two servers:   Server A (has SQL Server 2012 installed) Server B (has SQL Server 2014 installed)   Server A provides an API via IIS. The code both inserts and gets data from the database depending on the API call.  Server B does exactly the same for its local database.  I would like to have it so that the two databases are consistent copies of each other. I do believe this is what HA accomplishes.   So a load balancer, for example, might direct API calls to A sometimes, and B sometimes, but the data needs to be consistent either way.  I found some documentation on HA for SQL Server 2012, but not on how to do this for a mixed environment of 2012 and 2014.  Any help or pointers to documentation would be great! Thanks. Time Capsule cryptography? Does there exist any cryptographic algorithm which encrypts data in such a way that it can only be decrypted after a certain period of time?  The only idea that I can think of, is something like this: Seed a PRNG with a public value. Run the PRNG for a week and use the final value it produces to encrypt the message. Now anyone who runs that PRNG for a week starting from the seed value you made public can decrypt the message. Obviously this breaks down since they could use more computational power than you; if the time span was years then Moore's Law would apply, etc.  Is there anything like this other than physically burying a private key on a USB stick in a literal time capsule? I tried to create a script by echo'ing the contents into a file, instead of opening it with a editor  echo -e "#!/bin/bash \n /usr/bin/command args"  &gt; .scripts/command   The output:     bash: !/bin/bash: event not found   I've isolated this strange behavior to the .  $ echo ! !    $ echo "!" bash: !: event not found  $ echo \#! #!  $ echo \#!/bin/bash bash: !/bin/bash: event not found    Why is bang causing this?   What are these "events" that bash refers to?     How do I get past this problem and print "#!/bin/bash" to the screen or my file? "Images with street signs" CAPTCHA impossible to solve I just got challenged (over on ) by the new "Mark all images with street signs" CAPTCHA.  I must have tried a dozen times to solve it by checking all images that had street signs, but I never succeeded.  Of note is a text input field without instructions of what to enter there. I suspect that is the problem: It is expecting input, but doesn't tell me what to input.  It's either broken or I am missing something.  I am using Chrome on an iPhone in "full site" mode. At least two questions (, ) on this site indicate that in more than a few isolated cases at least, letters of recommendation are in fact written by the applicant and only signed off (or perhaps rephrased somewhat) by the recommender, who would actually be responsible for writing the letter.  Is there any evidence about the prevalence of this practice that goes beyond anecdotes? I realize that cross-country (cultural) difference may play a role. Delegates: Predicate vs. Action vs. Func Can someone provide a good explanation (hopefully with examples) of these 3 most important delegates:   Predicate Action Func Where can I get a clean ISO of a specific build of Windows 10? Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? Mass Uninstalling Just been working on a new laptop today (Acer) and it was pre-installed with all sorts of crapware. Mostly games and "backup utilities" and McAfee. I had to manually uninstall about 30 apps via Vista's Programs and Features  So it got me wondering is there any way of mass uninstalling or queuing an app to be uninstalled? Through Vista/XP you need to wait for Install Shield or whatever to close before beginning to uninstall the next app. Took way more time than it should have.  Just another one of those Windows annoyances! Any help appreciated as I often need to do this when I come across a slow machine that's just been clogged with a bunch of useless apps