Show that the sum of the next series is an irrational number $\sum_{1}^{\infty} \frac{1}{(n!)^2} $ Given:  $\sum_{1}^{\infty} \frac{1}{(n!)^2} $  development of the amount $1+ \frac{1}{4}+\frac{1}{36}+...$
Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any advanced/professional techniques. It can be proved using only basic techniques. My first attempt was to assume that the series does converge to a rational number $ \frac{p}{q}$ and then it follows by definition that for every positive epsilon there exists an integer k such that $ {\sum_{n=1}^{k} \frac{1}{(n!)^2} -\frac{p}{q}}&lt;=\epsilon$ I tried to get a contradiction but I failed.
Ok, after seeing , I have decided to skip to the end and go with .  So what I need is a beginner's practical guide to Git.  "Beginner" being defined as someone who knows how to handle their compiler, understands to some level what a  is, and has touched source control without understanding it very well.  "Practical" being defined as this person doesn't want to get into great detail regarding what Git is doing in the background, and doesn't even care (or know) that it's distributed.  Your answers might hint at the possibilities, but try to aim for the beginner that wants to keep a 'main' repository on a 'server' which is backed up and secure, and treat their local repository as merely a 'client' resource.  So:  Installation/Setup    How do you set up Git?  Try to cover Linux, Windows, Mac, think 'client/server' mindset.         Working with the code          How do you handle binary files (visio docs, for instance, or compiler environments)? How do you merge files changed at the "same time"?    Tagging, branching, releases, baselines    How do you pull a particular 'release'?   How do you resolve conflicts and complete the merge? How do you merge parts of one branch into another branch?        Other   Describe and link to a good GUI, IDE plugin, etc. that makes Git a non-command line resource, but please list its limitations as well as its good.    - Cross platform, included with Git  - Cross platform history viewer, included with Git  - Mac OS X  - Mac OS X history viewer  - Cross platform, commercial, beta  - console GUI for Linux  - GUI for Windows, Linux  - package for Windows, includes friendly GUI  Any other common tasks a beginner should know?     How do I work effectively with a subversion repository set as my source control source?   Other Git beginner's references                 Delving into Git     (and )    I will go through the entries from time to time and 'tidy' them up so they have a consistent look/feel and it's easy to scan the list - feel free to follow a simple "header - brief explanation - list of instructions - gotchas and extra info" template.  I'll also link to the entries from the bullet list above so it's easy to find them later. How can I move all the window controls to the right (or left)? Downloaded Ubuntu tweak and set window controls to the right side. Only some programs follow the rule. Pidgin as an example has window controls on the right. But Chrome, Thunderbird, Geany, still have controls on the left.  How can I move ALL window controls(close/min/max) to the right (or left)? AMPscript Error with XML I have a Partner sending some data to a Triggered Email/Data Extension which includes a column for XML data.   The trigger is erroring and not allowing the data to get to the Data Extension.   Below is what they are sending over. Is the formatting of the XML incorrect?  XML data that I've used in the past in Data Extensions starts with....   &lt;?xml version="1.0" encoding="UTF-8" standalone="yes"?&gt;   This version of XML starts with... {"XML":  Could that be it?  {"To": {"SubscriberKey": "test@acme.com", "ContactAttributes": {"SubscriberAttributes": **{"XML":** "&lt;root&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;dark chocolate nuts &amp; sea salt&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;15127&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/15127.html&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/nuts.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;crunchy peanut butter&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;41800&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/images/butter.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;11.37&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;9.55&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;pear cinnamon &amp; pecan&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;17227&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/apples.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;/root&gt;", "SHIPPING_LNAME": "Tester", "FIRST_NAME": "Test", "LAST_NAME": "Tester", "ORDER_TOTAL_SAVINGS": -5.97, "MERCHANT_ORDER_ID": "1810002352", "SHIPPING_ADDRESS_STATE": "NY", "ORDER_PLACE_DATE_LONG": "Fri 11 January 2019", "SHIPPING_ADDRESS_ZIP": "10004", "ORDER_SHIPPING": 0.00, "CARD_TYPE": "Visa", "ORDER_SUBTOTAL": 47.32, "ORDER_PLACE_DATE_SHORT": "01/11/2019", "SHIPPING_FNAME": "Test", "SHIPPING_ADDRESS_TOWN": "new york", "ORDER_TOTAL": 47.32, "CARD_EXPIRE": "12/2027", "CARD_HOLDER": "Test Tester", "SHIPPING_ADDRESS_1": "75 broad st", "SHIPPING_ADDRESS_2": "fl 23"}}, "Address": "acme@test.com"}}   AMPSCRIPT IN EMAIL Calling the XML....  %%[   var @LOOPINGAPPENDIX  set @LOOPINGAPPENDIX = AttributeValue("LOOPINGAPPENDIX")  set @productRows = BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM", 0)  ]%%   %%[   for @j = 1 to RowCount(@productRows) do       set @productXML = Concat("", Field(Row(@productRows, @j), "XML"), "")      set @ITEMPRODUCTNAME = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_NAME", 0), 1), "Value"))      set @ITEMPRODUCTSKU = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_SKU", 0), 1), "Value"))      set @ITEMPRODUCTURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_URL", 0), 1), "Value"))      set @ITEMPRODUCTIMGURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_IMG_URL", 0), 1), "Value"))      set @ITEMQTY = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_QTY", 0), 1), "Value"))      set @ITEMSUBSCRIPTIONPRICE = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_SUBSCRIPTION_PRICE", 0), 1), "Value"))      set @ITEMTOTAL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_TOTAL", 0), 1), "Value"))      set @rowColor = Iif(Mod(@j,2) == 0, "#efefef", "#ffffff")   ]%%   %%[endif]%%   %%[next @j]%% What are all the common undefined behaviours that a C++ programmer should know about?  Say, like:  a[i] = i++; How should a user's death be handled? It is a bit of a morbid concept, but having recently had a friend die unexpectedly and having to deal with the online repercussions of this event has gotten me to thinking about what should happen to my accounts when I kick the bucket.  I haven't had any great insights yet, but here are some things I think are worth thinking about:   an entry in the FAQ telling the person's loved ones what to do with respect to the account would be extremely helpful, specifically who to contact, how to contact them, and what proof is necessary some form of optional date of death on the user account screen some way to mark the user as no longer being able to generate content and blocking any attempt to login as the user.   The danger I see in implementing such features is the possibility of someone faking another person's death and thereby locking a user out of his or her account or worrying his or her friends. Is there a quick way to get to the Control Panel in Windows 8? Obviously in all previous versions of Windows you could simply click START and then select "Control Panel" to get to it, but on Windows 8 there is no such option.  Is there an easy way to quickly get to it without having to press WIN, type "Control Panel", and then press Enter? How do we know photons have spin 1? I've already see a  by Marko Riedel which I list it follows:     The standard way to treat these sums is to integrate   $$ f(z) = \frac{1}{(z+\alpha)^2} \pi \cot(\pi z)$$   along a contour consisting of a circle of radius $R$ and with $R$ going to infinity and hence being larger than $\alpha$, where the circle does not pass through the poles on the real axis.   Now along the semicircle in the upper half plane we have   $$|f(z)| \le \frac{1}{(R-|\alpha|)^2}\pi \left|\frac{e^{i\pi R\exp(i\theta)} + e^{-i\pi R\exp(i\theta)}} {e^{i\pi R\exp(i\theta)} - e^{-i\pi R\exp(i\theta)}}\right|= \frac{1}{(R-|\alpha|)^2} \pi  \left|\frac{e^{2i\pi R\exp(i\theta)}+1}{e^{2i\pi R\exp(i\theta)}-1}\right| &lt; \frac{1}{(R-|\alpha|)^2} \pi   \frac{1+e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}} {1-e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}}$$   This last term is clearly $O(1/R^2)$ as $R$ goes to infinity as the quotient of the two exponentials goes to one since $\exp(-R)$ vanishes and there is no singularity when $\theta = 0$ or $\theta = \pi$ as $R\cos\theta = \pm R$, which is not an integer by the assumption that the circle avoids the poles and hence cannot be one.     My question: It is clear that for each $z=Re^{i \theta}$ , $f(z)=O(1/R^2)$ ，but can this guarantee that the last term on RHS is uniform bounded ? Since $\theta$ varies on a compact interval $[0, \pi]$ , I want to show that for each $ \theta \in [0, \pi]$ , and a fixed positive number $M \gt 1 $ there exist an open ball contains $\theta$ and a fixed $R_{\theta}\gt 0$ , for every $R \ge R_{\theta}$ and $x \in $ the open ball $$\frac{1+e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} {1-e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} \le M$$  For $\theta \neq 0,\pi$ , it is easy to find the desired $R_{\theta}$ and the open ball , but if $\theta = 0$ or $ \theta = \pi$ , for every open ball containing $\theta$ , it behave erratically near $\theta$ , I have no idea how to deal with this . What inspired Tauriel's story arc? I know Tauriel is a Jackson created character, but I wondered if anywhere in the legendarium there was fodder for the story arc between her and Kili.  I do not have access to all my books for the time being and recall stories of love between men and elves in the Silmarillion and Appendices but don't remember any such stories between elves and dwarves and wondered if there was any basis of inspiration for this aspect of their relationship within Tolkien's works.   This movie, of course, departs even further from the original than those made before it, but as a story unto itself, I can still appreciate the ride, so if there is no Tolkien basis, what inspired Jackson, Boyens, and Walsh to create this particular arc?  Is there interview info or anything? How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$? Consider this limit:  $$ \lim_{x \to 1} \frac{\sqrt x -  1}{ \sqrt[3] x -  1}  $$  The answer is given to be 2 in the textbook.  Our math professor skipped this question telling us it is not in our syllabus, but how can it be solved? Transferring Lyx Keyboard Shortcuts I recently updated my version of Lyx from 1.6.7 to 2.0.6. I would like to transfer my old keyboard shortcuts to this new version, but I am having difficulty because I was inconsistent with the way that I set them up:   Some were defined in the specified .bind file,  but others were modified in the lyx preferences menu.   I found that when importing my old .bind file into this updated version of lyx, it imported the keyboard shortcuts that were in the .bind file, but not those that I had added in the preferences menu.  Does anyone know if these keyboard shortcuts that were added via the preferences menu are stored in another .bind file somewhere in the lyx directory? Has anyone else successfully transferred keyboard shortcuts that had been implimented in the preferences menu? Count number of occurences for each unique value Comparing values between two tables How to compare values between two different tables?  I have two tables as shown below:   Table 'A' sql_id       OPERATION       OPTIONS  63kf3an2j0pk SELECT STATEMENT 63kf3an2j0pkc TABLE ACCESS FULL  Table 'B' sql_id       OPERATION     OPTIONS 63kf3an2j0pk SELECT STATEMENT 63kf3an2j0pkc INDEX UNIQUE SCAN IDX_THU   I need to compare the values in Table 'A' and Table 'B' for the given sql_id. My question is how to find the changed values.  In this case in Table 'A' the second line is TABLE ACCESS FULL. Table 'B' the the second line is INDEX UNIQUE SCAN IDX_THU. So here the value has changed. How can I find that using query? Something like when matched then "matched" when not changed "value changed".  I'm looking for output something like below:     sql_id           comaparion outcome dmqb1hzrf5uxh   No change in the plan 74anujtt8zw4h   plan changed   Can I do something like this?  select sql_id,case when ( select sql_id,OPERATION,options  from  a where sql_id='dmqb1hzrf5uxh' minus  select sql_id,OPERATION,options from  b where sql_id='dmqb1hzrf5uxh') is null then 'matched'  else 'changed' end as status;   I am comparing the explain plans before and after patch. The table 'A' is before patch result of the sqlid and table 'B' is after patch result. Now I want to compare the plans in both the tables. I want to compare the plan by using a query. Can not start Chrome via command line I have a script that needs to run on multiple machines. The problem is, if I open a command line and type in chrome or chrome.exe, I get the same error message that it is not recognized as an internal or external command... I have already added it to my paths (as per screen shot below, see bottom entry in the paths) . It works this way on other computers.  The command prompt has been restarted (as has the computer) since updating the paths. Why does it not open Chrome as it does on my other machine(s)? Whilst entering C:\Program Files (x86)\Google\Chrome\Application\chrome.exe directly into the command line and pressing enter does make Chrome launch, I don't see why the path is not working. Automatically capitalise the first word in each sentence Getting a collection of index values using a LINQ query Is there a better way to do this?  string[] s = {"zero", "one", "two", "three", "four", "five"};  var x =  s .Select((a,i) =&gt; new {Value = a, Index = i}) .Where(b =&gt; b.Value.StartsWith("t")) .Select(c =&gt; c.Index);   i.e. I'm looking for a more efficient or more elegant way to get the positions of the items matching the criteria. I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers.  First, take the numbers between zero and one, exclusive.  The following sequence of real numbers is suggested so that we have bijection.    0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.10, 0.11, ... , 0.99, 0.001, 0.002, ... , 0.999, 0.0001, etc.  Obviously, this includes repeats, but this set is countable.  Therefore, the set of all numbers between zero and one is a subset of the above countable set, and is thus countable.  Then we simply extend this to all real numbers and all the whole numbers themselves, and since the real numbers, as demonstrated above, between any two whole numbers is countable, the real numbers are the union of countably many countable sets, and thus the real numbers are countable.    Please help me with this.  I understand the diagonalization argument by Cantor, but I am curious specifically about this proof which I thought of and its strengths and flaws.  Thanks. The + vowel letter I've been told that when "the" is proceeded by a vowel sound, like "apple" or "hour", it's pronounced as "thee" and not as "thu".  But after listening to a couple of songs, I noticed that sometimes this "rule" is not followed. Take for example the two Katy Perry's songs, "Roar" and "The one that got away". In the first she sings "I got thee eye of thu tiger", but in the second she sings "Thu one that got away".  I don't know if it was sang this way to better suit the song melody (I understand nothing about those techniques), but I got confused. What's the correct pronunciation?  Thanks in advance. I don't know so much about coordinate systems...  In my office we use to deal with spatial data coming from archaeological sites. Each site has its own x-y-z coordinate system (GCS). Three simple ortogonal cartesian axis. In the last years we have been managing this spatial data through GIS software (ArcGIS), without using specific coordinate system (just leave it as "undefined")  I'd like to know if there exisits any GCS designed to deal with such datasets using simple cartesian orthogonal axis, without grid distortions of the typical GCS. In addition, I'd like to know if this system is suitable for using it in an online mapping application.  By the way, we manage 2D (ArcMap) and 3D (ArcScene) environments and work with "mm" as length base unit.   If such a thing doesn't exists, maybe someone knows how to create it. Search text within multiple pdfs and docs I got a bunch of notes written by other students, but they are from an old textbook that didn't deal with everything in the same order, so I need to search through the notes for every chapter (each individual chapter is in a different .pdf or .doc) for "trace conditioning" for example.  I used to use Google Desktop for this, I have Launchy now and I told it to search pdfs, but it only searches the titles, not the content.  Thanks for any help.