Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges.  So I know that just because  $\sum{a_n}$ converges, do not mean I can say anything about the converges of its square root. So I know that I can prove that if  $\sum{\sqrt{a_n}}$ converges then $\sum{\frac{\sqrt{a_n}}{n}}$ converges by Abel's Test, but I do not know where to start for the case where  $\sum{\frac{\sqrt{a_n}}{n}}$ diverges
I need help to solve following problem from Rudin's Mathematical analysis book:    Convergence of the series $\sum a_n$  implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n&gt;0$  I tried to construct a suitable convergence sequence $b_n$ such that $\sum b_n$ converges and $a_n \leq b_n$ but, I am not able to find such sequence $b_n$ .  Thanks for the help and sugestions.
When you hit up a site, such as cnn.com, does the TCP/IP packet contain the MAC address of your network card?  Could cnn.com theoretically record my MAC address? Detecting pattern at the end of a line with grep How is arccos derived? Stupid question but I need to understand:  If $ \cos = \dfrac{\text{adjacent}}{\text{hypotenuse}} $  What is arcosine? $ \text{adjacent} \cdot \text{hypotenuse}$? Is this the same for arcsine and arctangent? How do I add a non-deformed end cap to a tube? I want to create some parts that I can reuse for models. I was trying to make a steam punkish exhaust pipe. I want to be able to make the pipe of any length, and as curvy as I like but I want to keep the end cap always undeformed. I approximated my results by hand placing the "crown" at the end of this bezier curve.  I tried to make an array modifier that included the crown as the end cap, but that results in the end cap deforming. We regularly get questions about wormholes on this site. See for example  and . Various wormhole solutions are known, of which my favourite is  because it's closest to what every schoolboy (including myself many decades ago) thinks of as the archetypal wormhole.  The trouble is that Visser has pulled the same trick as  of starting with the required (local) geometry and working out what stress-energy tensor is required to create it. So Visser can tell us that if we arrange exotic string along the edges of a cube the spacetime geometry will locally look like a wormhole, but we know nothing about what two regions of spacetime are connected.  My question is this: suppose I construct a Visser wormhole by starting in Minkowksi spacetime with arbitrarily low densities of exotic matter and gradually assembling them into the edges of a cube, how would the spacetime curvature evolve as I did so?  I'm guessing that I would end up with something like Wheeler's bag of gold spacetime. So even though I would locally have something that looked like a wormhole it wouldn't lead anywhere interesting - just to the inside of the bag. I'm also guessing that my question has no answer because it's too hard to do any remotely rigorous calculation. Still, if anyone does know of such calculations or can point me to references I would be most interested. When it comes to , I often read recommendations like "Use adb logcat to find some more details". Looking for a good way to backup all my apps including their data,  refers to adb backup and adb restore. In several places one can find ways to remotely do on an Android device with the use of , such as transferring files using adb push or adb pull, accessing the  via adb shell, and more. So ADB seems to be a good thing to have for an Android user.  But how to get this "ADB thingy" on my computer? Not being a developer, installing the entire  seems a bit overkill. Is there a more minimalistic approach available? I was trying to think but cannot figure it out. For instance, if the interaction is small, for instance limited to a windows, the observers in each universe will see that the other goes in reverse. But what if they can start a more meaningful interaction? (for instance, if one observer crosses the window and moves to the other universe, will anything change? why and how fast? (at least at the beginning his thinking will move in the same time arrow than in his universe, but the physical processes around him will still violate the second law? Filling NoData gaps in raster using ArcGIS Desktop? Is it possible to enable System Protection from the command prompt? Heartbleed: are services other than HTTPS affected? The OpenSSL 'heartbleed' vulnerability () affects webservers serving HTTPS. Other services also use OpenSSL. Are these services also vulnerable to heartbleed-like data leakage?  I'm thinking in particular of   sshd secure SMTP, IMAP etc -- dovecot, exim &amp; postfix VPN servers -- openvpn and friends   all of which, on my systems at least, are linked to the OpenSSL libraries. Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$ When you attempt to modify a file without having write permissions on it, you get an error:  &gt; touch /tmp/foo &amp;&amp; sudo chown root /tmp/foo &gt; echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Sudoing doesn't help, because it runs the command as root, but the shell handles redirecting stdout and opens the file as you anyway:  &gt; sudo echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Is there an easy way to redirect stdout to a file you don't have permission to write to, besides opening a shell as root and manipulating the file that way?  &gt; sudo su # echo test &gt; /tmp/foo Check if a class is derived from a generic class I have a generic class in my project with derived classes.  public class GenericClass&lt;T&gt; : GenericInterface&lt;T&gt; { }  public class Test : GenericClass&lt;SomeType&gt; { }   Is there any way to find out if a Type object is derived from GenericClass?  t.IsSubclassOf(typeof(GenericClass&lt;&gt;))   does not work. What is $48\div2(9+3)$? I have two questions for you guys. I really appreciate your help.  1.- Is there a way to rename an Availability Group?  2.- Is there a way to get the T-SQL statements in order to re create a Availability Group in case that I need to drop it and create it again?  If there's information that you want me to share, please let me know and I will provide it. Can I talk to the blue alpaca? Reese's boyfriend seems to constantly be asleep. When I try and talk to him, she warns me that he's been hard at work all night, but I can't actually enter the shop at night.   Is there any way to talk to him or any benefit for doing so? I've tried to talk to him daily for a week (thinking it was something akin to talking to Sable) but I've had no luck. Trying to come in after-hours doesn't work. Disable App Update Notifications on OSX? In many languages, including English, the most important verbs are irregular. Examples include:   to be  to do  to get  to go  to have  to make   The same applies (roughly) to many other languages I know about (Dutch, German, French, Spanish, Swedish) and presumably to many other languages too.  Is there any reason why these everyday verbs tend to be irregular? There was a great SF story I read some time ago (probably 25+ years ago) where there was a well known treasure trove, guarded by a killer robot that would ask all comers a series of difficult questions from all areas (math, physics, literature, history, etc.)  But everyone eventually failed and died... until the last person managed to make it past the robot:     By just answering gibberish to the robot.  Unfortunately, after collecting a ton of treasure, the robot casually asked him why he wanted it, he answered in a sensible way (something like "because it's valuable"), and the robot killed him, because he failed the final test.   Any ideas who wrote this?  What it was titled? How is light emitted by an incandescent lamp I am looking for better understanding of how light is produced in an incadescent lamp. More specifically: how is the kinetic energy of electrons converted to light?  Are we dealing with interband transitions or with intraband relaxation involving photons? Is this Bremsstrahlung (electrons lose their energy as light when colliding with crystal impurities/defects)? Or is this a thetmal radiation resulting from Joule heating? How is the emission affected by presence of impurities and imperfections of the crystal lattice? Do phonons play a role? What properties make a material more suitable for use as a filament: should it be a metal? Should it have a crystalline structure? Will any metal produce light, if a high current is passed through it in vacuum?  Update The term describing the processes in the incadescent lamp is , see the posts on this subject  and .