Finding the limit of the sequence $(1 - 1/\sqrt 2) \dotsm ( 1 - 1/\sqrt {n+1})$
How can I find $\lim_{n\to \infty} a_n$
Is it coherent to extend $\mathbb{R}$ with a reciprocal of $0$? I'm sure this is an obvious question, but I'm having trouble finding the right words to type into Google.  I know that the definition of a ring allows that the additive identity not have a multiplicative inverse, but is this a requirement?  Specifically, is something like $\mathbb{R}\!\left[\frac{1}{0}\right]$ such that $\frac{1}{0} \cdot 0 = 1$ a ring, or does some contradiction arise from allowing the additive identity to have a multiplicative inverse? How do you empty the buffers and cache on a Linux system? In what sense is a scalar field observable in QFT? Consider a QFT consisting of a single, hermitian scalar field $\Phi$ on spacetime (say $\mathbb R^{3,1}$ for simplicity).  At each point $x$ in spacetime, $\Phi(x)$ is an observable in the sense that it is a hermitian operator (operator-valued distribution) on the Hilbert space of the theory, but is each such operator observable in a stronger, more physical sense?  Is there an experiment one could hypothetically perform to measure the value of such a field at a given spacetime point? This is one of those questions I glossed over while learning QFT, but now it's bugging me.  In particular, I think this point is central in preventing me from understanding certain basic assumptions in QFT such as microcausality which I also never really think about anymore. Significance of the second focus in elliptical orbits 1.In classical mechanics, using Newton's laws, the ellipticity of orbits is derived. It is also said that the center of mass is at one of the foci.   2.Each body will orbit the center of the mass of the system.  My question is : Are the assumptions in 1 and 2 correct?   Follow up question : Assuming the distance from the centre of the mass to each body remains the same, do we have two bodies orbiting the centre of the mass of the system in an elliptical or circular orbit?   Finally : With elliptical orbits, if the heavier mass is supposed to be in one of the foci, if there is any significance to second focus, what is it? Is it a Lagrange point by any chance or does it have some other mathematical property? How to concatenate text from multiple rows into a single text string in SQL server? Cleaning polygon after using dissolve to repair geometry in QGIS? I'm trying to obtain the boundaries of shapefile, thus to clean the inside as a matter of speaking. While I had expected that dissolve while get me a proper polygon, unfortunately it still has some remainders. How could I get rid of them?  Does QGIS have a tool similar to Repair Geometry?  Here's the source    Result below: Choice of ECC curve on USB token I would like to get a USB token to secure my keys. My use case is protection of 3 GnuPG keys that I will be using 10 times per day at least. I plan to create a new key ring from scratch. Because ECC seems more future-oriented than RSA, this is what I chose to use. I'm wondering which USB token to choose as well as which curve.  On  it is said that many people think curves recommended by the NIST or defined by Brainpool have a doubtful origin; therefore, they recommend the non-standardized Bernstein’s Curve 25519. On  one poster says that    He is not aware of any profound criticism of Brainpool curves, and Bernstein’s Curve 25519 is hard to protect against side-channel attacks when implemented in embedded devices.   As a result, I'm a bit lost as to which key/curve to choose. Is using "fruits" as the plural of "fruit" acceptable? How do I use SSMS to compare Database 1 Table 1 against Database 2 Table 1? I have two databases - Database1 and Database2.  Both databases contain a table that has similar structure, exemplified as follows:   =========================================================================================== | ID | Name | PhoneNoFormat | DialingCountryCode | InternationalDialingCode | InternetTLD | =========================================================================================== |    |      |               |                    |                          |             | ===========================================================================================   However, due to some reason, (a) one of the tables in one of the databases has data that is not exactly the same as (b) that contained in the other table in the another database.  So, how can I compare Database1.Table1 against Database2.Table1?  I tried using the following query, but nothing happened so was wondering if I have to rewrite it:  SELECT MIN(TableName) as TableName,        ID,        Name,        PhoneNoFormat,        DialingCountryCode,        InternationalDialingCode,        InternetTLD  FROM  (    SELECT  'Table A' as TableName,           A.ID,           A.Name,           A.PhoneNoFormat,           A.DialingCountryCode,           A.InternationalDialingCode,           A.InternetTLD    FROM [D:\DATABASE1.MDF].[dbo].[Table1] AS A    UNION ALL    SELECT 'Table B' as TableName,          B.ID, B.Name,          B.PhoneNoFormat,          B.DialingCountryCode,          B.InternationalDialingCode,          B.InternetTLD    FROM [D:\DATABASE2.MDF].[dbo].[Table1] AS B  ) tmp  GROUP BY ID, Name, PhoneNoFormat, DialingCountryCode, InternationalDialingCode, InternetTLD  HAVING COUNT(*) = 1  ORDER BY ID Galois group of irreducible cubic equation I am trying to buy a laptop, and I have seen in many place that they show GDDR3 and sometimes  DDR3. What is the difference between them?  afaik DDR3 is a type of RAM used for system memory, while GDDR3 is a type of RAM used for graphics cards. So does "G" only stand for Graphics or does it have another particular meaning? Proving a simple graph is a connected graph Does any proof exist that a simple graph with $n$ vertices such that the least vertex degree is $\geq \frac{n-1}{2}$ is a connected graph? (i.e. does such a proof have a name?) Example of $f,g: [0,1]\to[0,1]$ and Riemann-integrable, but $g\circ f$ is not? I left my cast iron pot on the gas stove, on high flame, and then forgot about it for 3 hours. It was empty and I was drying it before I oiled it to store it. After I discovered it and let it cool, I oiled it with a paper towel and all this rusty stuff came off and continues to come off. Is my pan ruined? What should I do? Is there any difference between "word-for-word translation" and "word-by-word translation" and is the latter actually valid? First off, some data: According to COCA word-for-word has 60 usages, 3 of them are &quot;word-for-word translation&quot;. Word-by-word has 26 usages, none of them are &quot;word-by-word translation&quot; (but some with &quot;transcription&quot;). The definition of word-for-word:  : In exactly the same or, when translated, exactly equivalent words : being in or following the exact words, verbatim : one word at a time, without regard for the sense of the whole  Only the last dictionary contains a definition for word-by-word, too:  : one word at a time  The definitions given by The Free Dictionary are, obviously, identical to each other. Google hits: Word-for-word ~21m Word-for-word translation ~318k Word-by-word ~3.8m Word-by-word translation ~95k According to usages and dictionaries word-by-word is, at least, less popular. And assuming that there may be a lot of usages from non-natives among the Google hits, this could be an indicator for word-by-word being even utterly wrong. In another forum I found the following :  When I translate something &quot;literally,&quot; (wörtlich) it still follows the main rules of the language I'm translating into. What you mean is &quot;word-by-word&quot; (wortwörtlich) to me.  I assume that this was written by a German but I don't know it. However, if this would be true a &quot;word-by-word translation&quot; would be a translation where I keep, for instance, the order of the words, disregarding if it makes sense in the target language. Some examples:  Original: word-by-word Word-by-word translation: Wort bei Wort (That's a terrible translation!) Original: It is critical to know... Word-by-word translation: Es ist kritisch zu wissen... (That's a terrible translation!) Original: Ich glaub, ich spinne. Word-by-word translation: I think I spider. (I guess only Germans understand this.)  A &quot;word-for-word translation&quot;, however, would be an attempt to keep the word-choice as close as possible but following the rules of the target language (e.g. order of words) and also considering if the statement still makes sense in the other language. Here are better translations for the examples above:  Wort für Wort Es ist wichtig zu wissen... I think, I'm going nuts. (Actually, this is not a word-for-word translation but rather a .)  So, my questions again:  As neither Oxford nor Merriam-Webster have any entries for word-by-word in their dictionaries: is word-by-word actually valid? If yes, is there any difference between &quot;word-by-word translation&quot; and &quot;word-for-word translation&quot;? If yes again, what is it specifically? How to find out where an error occurs? Copying prod multi-site installation to dev multi-site installation I am attempting to upgrade my Drupal 8.9.2 multi-site installation to D9. I was hoping to work this all out on a development installation, that is located in a subdirectory of my home directory on my NameCheap Shared Hosting plan. When I first installed this installation, I did so using the tarball method from years ago. I then followed the instructions given here (). I was able to get the current running prod version converted to use composer with out an issue. Before I go and start messing with the files, and attempting to upgrade to D9 through composer on a prod installation, I wanted to copy these files over to another installation where I can use it as a sandbox. To do this I:  copied the files on my server to a new directory made clone databases for the new installation truncated all the cache tables in the cloned DB changed all the database settings in the settings files to point to the cloned databases  I've done this before without any issues, and now I seem to be getting a WSOD error on all the sites running from the new installation. The error I'm getting is  PHP Fatal error:  Uncaught Error: Class 'Drupal\Core\Cache\DatabaseBackend' not found in ~/{new installation dir}/public_html/index.php:16.  If anyone has any thoughts what's going on I am all ears. Thank you in advance ArcPy Utility Network Trace Not Returning Any Result Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$. $Q$ is unitary, so $QQ^*=I$ If $A$ and $B$ are positive-definite, than $A=A^*$ and $B=B^*$. $A^*=(BQ)^*=Q^*B^*$ $A^2=AA=AA^*=(BQ)(Q^*B^*)=B(QQ^*)B^*=BIB^*=B^2$ $A^2=B^2$. I don't know how to use the fact that $A$ and $B$ are positive-definite to finish the proof. How do I pack textures for sending off to a render farm Whenever i try to send a project off to a render farm such as sheepit, it says it needs the textures packing with the file in order to render correctly.  How would I do this?