$$\sum_{k=0}^{\infty} \frac{(-1)^k(2k+1)}{(2k+1)^2 + 1} = \frac{\pi}{4}\mathrm{sech}(\frac{\pi}{2})$$ I tried to consider breaking up the alternating factor of the series by decomposing it into 2 separate series - one adding the even terms and the other subtracting the odd terms, but then I couldn’t figure out where to go. My attempt, letting $S$ be the sum of interest. $$S = \sum_{k=0}^{\infty} \frac{(2(2k)+1)}{(2(2k)+1)^2 + 1} -   \frac{(2(2k+1)+1)}{(2(2k+1)+1)^2 + 1}$$ Don’t know what else to do.
Solve $$\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$$ without complex integration.  This integral can be very easily solved with contour integration, but how can you solve it without taking a tour in the complex plane?
Tentative definitions in C and linking Badge overflow on users with too many badges When a user has &quot;too much&quot; reputation and &quot;too many&quot; badges, the design of the user card breaks:    The badge counters overflowing to another line. Can anything be done about this? For example, expanding the rep and badges to their own lines:  NameRepBadges Stack Overflow has successfully borrowed at least one XBox feature (Achievements).  I'd like to see another feature borrowed as well. Let's say I'm playing Halo online and another player is being a bigot.  I have the option to add him to a list of ignored users.  After doing that, all the racist things he says will never be heard through my headphones anymore. Now, on Stack Overflow, I'd like the ability to add a user to a list of ignored users.  This would hide all of that user's comments, and maybe his questions and answers as well (not sure if hiding questions/answers is necessary). This serves multiple purposes:  It would make it easy for users to avoid other users' flamebait. It would give users a more appropriate response to flamebait. The ability to start flame wars would be reduced, since fewer users would see those comments. Users would now have an incentive to be more diplomatic, as there is now a negative consequence to the behavior (the silent treatment). Administrators could review which users are the most often ignored, which would be a strong indication that someone should be  or even locked out of their account. Administrators wouldn't have to spend as much time dealing with complaints about abusive users, since users would have a better way to respond.    makes a good point in a comment to an answer, that I thought should get more visibility:  I had a problem, reported it and got the response &quot;not bad enough to do anything about, sorry&quot;. Which is kind of true, because the only options available to the mods are the nuclear one (penalty box) and the zero-effect one (do nothing). We need a middle ground option where abuse can be handled for you without also being handled for everyone. How to extend USB beyond the recommended maximum cable length of 5m? I'm planning to use USB port for an application(from a device to a PC) but the distance between the device and the PC will be around 15m. The recommended maximum USB cable length is 5m.  How can I buffer the data without loosing any information, safely? Would a USB hub work? How to adjust the size and placement of chapter heading in report class How could I adjust the size and placement of chapter heading in my custom class? I am using report class as my base. When I used titlesec package inside my custom class, it threw error     ! Package titlesec Error: Not allowed in `easy'   settings. Should I file Form 1040 or Form 1040 NR? I am currently on H1 B (since October 1, 2017). Last year when on F1 OPT, I had filed Form 1040 NR for my taxes. Which form is advisable for 2017, considering I am on H1 B now. What options do I have? Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$ I came across the following problem in my self-study of real analysis:     For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = \frac{1}{2}(a+b-|a-b|)$$   So $a \geq b$ iff $a-b \ge0$ and $b \ge a$ iff $b-a \ge 0$. At first glance, it seems like an average of distances. For the first case, go to the point $a+b$, add $|a-b|$ and divide by $2$. Similarly with the second case.   Would you just break it up in cases and verify the formulas? Or do you actually need to come up with the formulas? How do I properly access Windows software RAID 0? I am a new Ubuntu user with a fresh installation, and am eager to get started with it. However, all of my important data is on Windows software RAID 0 array, and I need it to stay there as I'm also dual booting Windows. I need access to this data on Ubuntu, and can't really get anything real done until I have access.  I have already researched this subject as best as I could, and I managed to find a very helpful post:      The post describes how to get Ubuntu to see a Windows RAID 0 array that is made up of two drives. The main command used is sudo mdadm --build /dev/md0 --chunk=64 --level=0 --raid-devices=2 /dev/sdd2 /dev/sdc2. According to that user, and the other posters in the thread, it actually does work. That's great.  I have not tried following these instructions yet. Why? The post includes a warning about how you must not write to it if you enter the wrong chunk size; it's understandable how that could cause problems. My concern is that my setup is different to their example, and I am not sure that the commands should be entered exactly the same for my setup. I am afraid to break it by doing it wrong, and therefore wish to get the advice of somebody more experienced.  This is how my setup differs from their example:   I have three 1 TB drives, not two drives (of whatever size they used). I have two RAID 0 partitions spread across those three drives: one 500 GB and 2.3 TB. This means that I need to NOT use the full disks when creating the RAID array, but instead use just part of them. I used a non-default block size for at least one of my RAID 0 partitions when I set them up years ago. I have no idea if this block size is the same as the chunk size that they mention. My 500 GB partition has a block size of 4 kb (4096 bytes per cluster), and my 2.3 TB partition has a block size of 64 kb (65536 bytes per cluster).   The relevant output from sudo lsblk -o NAME,FSTYPE,SIZE,MOUNTPOINT,LABEL (for just those three RAID0 drives) is:  NAME   FSTYPE   SIZE MOUNTPOINT            LABEL sdb           931.5G             ├─sdb1            1M             ├─sdb2          127M             └─sdb3        931.4G             sdc           931.5G             ├─sdc1            1M             ├─sdc2          127M             └─sdc3        931.4G             sdd           931.5G             ├─sdd1        166.7G             └─sdd2        764.7G    The relevant output from cat /proc/partitions (for just those three RAID0 drives) is:  8       16  976762584 sdb 8       17       1024 sdb1 8       18     130048 sdb2 8       19  976631478 sdb3 8       32  976762584 sdc 8       33       1024 sdc1 8       34     130048 sdc2 8       35  976631478 sdc3 8       48  976762584 sdd 8       49  174763008 sdd1 8       50  801865728 sdd2   The "Disks" program in Ubuntu displays the following partitions for my drives:  /dev/sdb:   GUID Partition Table . /dev/sdb1:  1.0 MB, Microsoft LDM metadata. /dev/sdb2:  133 MB, Microsoft Reserved. /dev/sdb3:  1.0 TB, Microsoft LDM data. /dev/sdc:   GUID Partition Table partitioning. /dev/sdc1:  1.0 MB, Microsoft LDM metadata. /dev/sdc2:  133 MB, Microsoft Reserved. /dev/sdc3:  1.0 TB, Microsoft LDM data. /dev/sdd:   Master Boot Record partitioning. /dev/sdd1:  179 GB, Unknown. /dev/sdd2:  821 GB, Unknown. /dev/sdd:   136 MB, Unallocated space.   I hope I've provided enough information here. So now, my question is this: what is the proper command for me to enter with my setup, so that I can access both of my Windows RAID 0 partitions from Ubuntu?  Thank you very much in advance. Complex towers: $i^{i^{i^{...}}}$ How to resolve a Java Rounding Double issue How to programmatically click a button in WPF? Convert a string to an enum in C# What's the best way to convert a string to an enumeration value in C#? I have an HTML select tag containing the values of an enumeration. When the page is posted, I want to pick up the value (which will be in the form of a string) and convert it to the corresponding enumeration value. In an ideal world, I could do something like this: StatusEnum MyStatus = StatusEnum.Parse(&quot;Active&quot;);  but that isn't a valid code. Search/browse my own questions easily No matter how nice the composition, flow, and lines of my design are, if I am not copying directly from nature, I have a very difficult time picking a decent color scheme.  It either looks too drab, or too garish.  Is there a good resource that will help me to choose a color scheme that has 2 or 3 colors, and the colors do not clash?  I'm not looking for specific colors or color schemes.  What I need is a resource that will help me choose a color (or colors) that go nicely with, or provide a good accent to, whatever main color I am working with on the design. Altering layer's CRS to correct Layer Misalignment in QGIS I've got a vector layer in my project which isn't aligned with the rest of the project (see attached JPEG).  These are work files so I'm not sure when they were created or how they might have been messed with since they were created.  We moved from ArcGIS to QGIS a couple of years ago.  The infilled blue vector layer is the one I need to move, slightly north and east, so that it correlates to the blue line (also a vector).  All layers (and the project) are currently in EPSG:27700 - OSGB 1936.   If I need to create a new CRS based on OSGB 1936 and give it some new parameters, then how would I go about this?  Further information:  This is on Exmoor in SW England, and the data is uniformly almost exactly 8m out to the SW point to point - i.e I need to shift it 8m NE. Is there a way to copy maintenance plans from one SQL Server 2012 to another? Both servers are not accessible via the same copy of Management Studio at the same time (on different networks). How do I watch a file for changes? I have a log file being written by another process which I want to watch for changes. Each time a change occurs I'd like to read the new data in to do some processing on it.  What's the best way to do this? I was hoping there'd be some sort of hook from the PyWin32 library. I've found the win32file.FindNextChangeNotification function but have no idea how to ask it to watch a specific file.  If anyone's done anything like this I'd be really grateful to hear how...  [Edit] I should have mentioned that I was after a solution that doesn't require polling.  [Edit] Curses! It seems this doesn't work over a mapped network drive. I'm guessing windows doesn't 'hear' any updates to the file the way it does on a local disk. How can anything ever fall into a black hole as seen from an outside observer? The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? What is the Unix command to create a hardlink to a directory in OS X? How do you create a hardlink (as opposed to a symlink or a Mac OS alias) in OS X that points to a directory? I already know the command "ln target destination" but that only works when the target is a file. I know that Mac OS, unlike other Unix environments, does allow hardlinking to folders (this is used for Time Machine, for example) but I don't know how to do it myself. GPU Rendering causing strange lines When I render using my r9 390 graphics card, weird lines are appearing over the render. This problem has just started happening recently, as a few days ago I was rendering the same scene without the lines appearing. I don't believe I've changed any settings in Blender and the below scene is from a new file, so render settings should be default.   Rendering using my CPU is fine:   The only thing I can think that has changed is I recently updated my card to the latest drivers. Playing games with it is still fine and the card is quite new so I wouldn't think it's a problem with the card itself.