How do you compute $\prod_{n=2}^\infty\bigg(1-\frac1{n^2}\bigg)$   I have no clue on how to start. Any help would be appreciated.
While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$?
When a PHP application makes a database connection it of course generally needs to pass a login and password. If I'm using a single, minimum-permission login for my application, then the PHP needs to know that login and password somewhere. What is the best way to secure that password? It seems like just writing it in the PHP code isn't a good idea. Is there a reason behind the ordering of letters in the English alphabet? Is there a reason behind the ordering of letters in the English alphabet? i.e. why are we taught “A,B,C,D,E,F,...,Z”? Why not “L,A,S,U,I,Z,...,C”?  I am asking this because, in some of the languages I know, I am told that the ordering of the letters in the alphabet is based on the ease with which they can be pronounced or the frequency with which they are used or depending on the part of the vocal cord that needs to be stressed to pronounce the letter.  Is there a similar rationale?  EDIT  You can see a couple of references for arrangement of letters in Sanskrit  and . How do you format the day of the month to say "11th", "21st" or "23rd" (ordinal indicator)? Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ I've been looking at  $$\int\limits_0^\infty  {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$  It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:  $$\displaystyle \int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^3}}}dx = } \frac{\pi }{3}\frac{1}{{\sin \frac{\pi }{3}}} = \frac{{2\pi }}{{3\sqrt 3 }}$$  $$\int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^4}}}dx = } \frac{\pi }{4}$$  $$\int\limits_0^\infty  {\frac{{{x^2}}}{{1 + {x^5}}}dx = } \frac{\pi }{5}\frac{1}{{\sin \frac{{2\pi }}{5}}}$$  So I guess there must be a closed form - the use of $\Gamma(x)\Gamma(1-x)$ first comess to my mind because of the $\dfrac{{\pi x}}{{\sin \pi x}}$ appearing. Note that the arguments are always the ratio of the exponents, like $\dfrac{1}{4}$, $\dfrac{1}{3}$ and $\dfrac{2}{5}$. Is there any way of finding it? I'll work on it and update with any ideas.    UPDATE:  The integral reduces to finding   $$\int\limits_{ - \infty }^\infty  {\frac{{{e^{a t}}}}{{{e^t} + 1}}dt} $$  With $a =\dfrac{n+1}{m}$ which converges only if   $$0 &lt; a &lt; 1$$  Using series I find the solution is     $$\sum\limits_{k =  - \infty }^\infty  {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}}} $$ Calculate mean of ordinal variable I've read in a number of places that calculating the mean of an ordinal variable is inappropriate. I'm trying to get an intuition for why it might be inappropriate. I think it is because, in general, an ordinal variable is not normally distributed and so calculating the mean will give an inaccurate representation. Could somebody give more detailed reasoning for why calculating the mean of an ordinal variable might be inappropriate? When does HashSet 'add' method calls equals? Charts from numbers not dynamically updating in Pages/Keynote I have the 2015 versions of Pages, Numbers, and Keynote on Yosemite.  I created a chart in numbers and then copied it to Pages or Keynote. I then want to be able to change the data in Numbers and see the chart updated in Pages or Keyonte. However, I don't see an option to "refresh".  Is there a way to dynamically link Numbers to Pages or Keynote? Deleting a suspicious malware (imklaunchagent) in Mac I have a MacBook Pro (macOS Mojave) laptop and have recently noticed a suspicious process in Activity Monitor with this name : imklaunchagent    I checked this folder: /System/Library/Frameworks/InputMethodKit.framework/Versions/A/Resources/imklaunchagent and it seems imklaunchagent and its related files cannot be removed. I tried to remove it via Terminal and this message showed up: "Operation not permitted". However, in "Security and Privacy" of "System Preferences", I chose Terminal to have full disk access, still can't delete the file and the same message appears.  I installed the free version of Combo Cleaner and it detected other viruses, none of which are imklaunchagent. Is imklaunchagent a malware? How can it be deleted? How do I control how Emacs makes backup files? Emacs puts backup files named foo~ everywhere and I don't like having to remember to delete them.  Also, if I edit a file that has a hard link somewhere else in the file system, the hard link points to the backup when I'm done editing, and that's confusing and awful.  How can I either eliminate these backup files, or have them go somewhere other than the same directory? Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$ bash if not multiple conditions without subshell? I want to combine multiple conditions in a shell if statement, and negate the combination.  I have the following working code for a simple combination of conditions:  if [ -f file1 ] &amp;&amp; [ -f file2 ] &amp;&amp; [ -f file3 ] ; then   # do stuff with the files fi   This works fine.  If I want to negate it, I can use the following working code:  if ! ( [ -f file1 ] &amp;&amp; [ -f file2 ] &amp;&amp; [ -f file3 ] ) ; then   echo "Error: You done goofed."   exit 1 fi # do stuff with the files   This also works as expected.  However, it occurs to me that I don't know what the parentheses are actually doing there.  I want to use them just for grouping, but is it actually spawning a subshell?  (How can I tell?)  If so, is there a way to group the conditions without spawning a subshell? How to hide empty (value 0) ybars with pgfplots? I installed Dev C++ using Wine on Ubuntu 18.04 and the program is compiling but it not showing any output screen. Why is the QGIS map composer grid showing unreasonable lat/lon? I need to print a map with coordinates in the format of Degree, Minute and second. CRS is set to WGS 84 (EPSG 3857) for both layers and on the project properties it is set to enable transformation on the fly. But then, when in map composer I get the following coordinates, which make no sense to me. I know it is not recommended, but is it at all possible to pass the user's password to scp?  I'd like to copy a file via scp as part of a batch job and the receiving server does, of course, need a password and, no, I cannot easily change that to key-based authentication. Right now I'm using echo &quot;Hello World&quot; &gt;&gt; file.txt  to append some text to a file but I also need to add text below a certain string let's say [option], is it possible with sed? EG: Input file Some text Random [option] Some stuff  Output file Some text Random [option] *inserted text* Some stuff I'm going to mount my 47" LCD above my fireplace. I'm not looking for arguments against this decision. I'm looking for wiring solutions. The diagram below should help explain the situation.   _________________________________________________ |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | | p |   | m | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    #################    |     |     | |     |     |    ##             ##    |     |     | |     |     |    ##             ##    |     |     | |    P|     |    ##             ##  m |     |P C  | |_____|_____|____##_____________##____|_____|_____|    | = stud # = fireplace brick P = existing power outlet C = existing cable outlet p = new power outlet m = new media outlet   Essentially, there is an existing power outlet on either side of the fireplace and a cable outlet on the far right side of the fireplace. I'm intending to put my non-TV media equipment on the right side of the wall and the television above the fireplace. This means I need two things above the fireplace: power and HDMI. I don't want to just run an HDMI cable in the wall. I'd prefer to have some sort of conduit to simplify replacing the cable or having a second (with one of these on each end: ). I only need to drill through two studs that are 6" apart above the right side of the fireplace in order to access the vertical space where I want the other end. For the power, I need to drill through two 6"-spaced studs above the left-side of the fireplace, but then another 2 full-height studs.  There are a couple of caveats:   This is an externally facing wall, which means there's insulation. I'm assuming this will make things more difficult. I'd like to refrain from cutting more holes in the wall than is necessary for the outlets (to reduce drywall damage/repair).   I've considered a couple of options:   Cut the openings for the outlets above the fireplace, and use a flexible drill bit to drill through the studs above the fireplace. This minimizes dry-wall damage, but how difficult is this with insulation in the wall, how can I feed the conduit through those holes, and how would I drill through the full-height studs on the left for power? Cut a wide section of dry-wall (4" high?) from the desired media outlet to the right until reaching the gap above the desired lower outlet. This results in some dry-wall damage, but would allow more control for drilling holes in the studs and running conduit. This still doesn't solve the problem of getting through the full-height stud for the power. For the power, I would need to extend the cut-out two studs further.   What should I do? How does gravity work on Serenity? When the good ship Serenity is traveling in deep space, the crew experiences what looks like an Earth-normal (or Earth That Was-normal) level of gravity. Is it ever explained how the ship is able to generate artificial gravity? The ship doesn't spin, so  is out.  is a possibility, but I think the gravity in the ship pulls in a direction that is perpendicular to the direction the ship is usually traveling in, not opposite, like this method would require.   So: how does she pull it off?   Extra credit: how does she pull it off in the episode ? Serenity's engine is totally dead, to the extent where not even life support systems are online. Yet Mal still has to valiantly haul himself around on the floor to replace the compression coil, so whatever creates the artificial gravity on Serenity is clearly still going. How? How to jusify {e,n} is a valid RSA public key? What does // mean in a path?