Statement for Query
IF statement inside a LINQ SELECT
prove $\lim_{n\to\infty}{\frac{a^n}{n!}}=0$ I have the proof but i don't understand one part. The proof (for $a&gt;1$) goes as follows:  there exists $k \in N$ such that$a&lt;k, \frac{a}{k}&lt;1$ and since $\frac{a}{k+i}&lt;\frac{a}{k}, i \in N$ $$\frac{a^n}{n!}=\frac{a}{1} \cdot \frac{a}{2} \cdot...\cdot\frac{a}{k}\cdot\frac{a}{k+1}\cdot...\cdot\frac{a}{n}&lt;\frac{a^k}{k!}(\frac{a}{k})^{n-k}=\frac{a^k}{k!}\cdot \frac{a^na^{-k}}{k^nk^{-k}}$$  $$0&lt;\frac{a^n}{n!}&lt;\frac{k^k}{k!}(\frac{a}{k})^n$$  then $$0 \leq \lim_{n\to\infty}{\frac{a^n}{n!}}\le\frac{k^k}{k!}\lim_{n\to\infty}{(\frac{a}{k})^n}$$  Now, the part I don't understand is in the expansion of $\frac{a^n}{n!}$. How does one come with $&lt;\frac{a^k}{k!}(\frac{a}{k})^{n-k}$ part, to be more precise, how does one come up with $(\frac{a}{k})^{n-k}$ part of the inequality? For PhD applications, I have mentioned that I am interested in some research areas. Besides, I am thinking about mentioning some professors' names, so that it shows I have read their websites and I am not just sending out generic statements applicable to many schools.  But is it good to mention some professors' names in the statement of purpose, if I don't know them personally? In what cases is it good, and in what cases it isn't?  For example, is it good to only mention    those whose research directions are aligned with my research interests, or/and those whose books or notes I have heard of, or better, might have read some parts of, or/and those who are already established and famous (such as having certain titles), or/and those that are directing the PhD graduate programs, or/and those that are heads of the departments?    What will other professors who are not mentioned think? How will you do, if it were you? Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$? Let $M$ be a finite extension of $E$ and let $E$ be a finite extension of $F$. Then $M$ is a finite extension of $F$ and $[M:F] = [M:E][E:F]$.   Is there an easy explanation and/or proof for this theorem? My instructor gave an incomplete proof in class but got stuck and I'm kind of confused of what's going on. Allow the owner of a post to approve a suggested edit even if it was rejected by reviewers What is Interface method in salesforce Problem in notation: $(\vec{v}\cdot\vec{\nabla})\vec{v}$ versus $L_v(\vec{v})$ How to use unnumbered chapters with KOMA-script? I am having some issues with the hyperref package. I would like to create a Introduction chapter without a number so I use \chapter*{Introduction} and \addcontentsline{toc}{chapter}{Introduction} because I want it in the table of contents (see example).  But then the link in my PDF viewer gets it wrong and also the automatically created header is no longer correct (see picture). The table of contents shows the right page number (p.1) but the link is also wrong.   What is my mistake here?  \documentclass{scrbook}  \usepackage{hyperref} \usepackage{blindtext} \usepackage{glossaries}   \makeglossaries  \newacronym{eg}{e.g.}{% for example }   \begin{document}   \frontmatter  % abstract before the table of contents \chapter*{Abstract} \addcontentsline{toc}{chapter}{Abstract} \blindtext  \tableofcontents  % chapter with acronyms \cleardoublepage \phantomsection  \printglossary[type=\acronymtype] \addcontentsline{toc}{chapter}{Glossary}  \cleardoublepage \mainmatter   \phantomsection % \chapter{Introduction} % with this everything is correct but I don't want the number \chapter*{Introduction} \addcontentsline{toc}{chapter}{Introduction}  \blindtext  \pagebreak  \blindtext  % first chapter \chapter{Chapter1}  \gls{eg} \blindtext  \pagebreak  \blindtext   \end{document} How to avoid overflow using \IEEEpubid on the first column? I am trying to write a journal paper using the IEEEtran.cls template.  My problem is that the Publication ID Mark overflows into my thanks section.  This is the bottom of the title page:    From the documentation of IEEEtran.cls we see that:    I included the command \IEEEpubidadjcol into the second column. This prevents the overflow in the second column as you can see in my previous image, but lets the first column run freely.  I tried including \IEEEpubidadjcol into the first column aswell, but that causes compilation errors.  Please, could you give me some hint on how to solve this problem without doing something illegal with respect to the IEEE modus operandi?  EDIT:  As the answer suggests, the problem can be solved using the \IEEEpubidadjcol command twice. When I tried this, I got an error:  Infinite glue shrinkage found in box being split   In spite of this error a correct pdf is produced. The error goes away when I do not include packages fixltx2e and dblfloatfix. Alternative to Adobe Illustrator Can you recommend a good alternative to Adobe Illustrator?   I am learning vector based graphic design, but I am not ready to make such a large investment as to purchase a copy of Adobe Illustrator. Also, Illustrator might be a little more power than I need at the moment. I am using a Macintosh. Calculate the area on a sphere of the intersection of two spherical caps How can I maximize the "blurry background, sharp subject" (bokeh) effect? I know that this effect occurs when there's a shallow depth of field.  My question is, what are the various ways I can increase this effect in my photos when I'm taking them?    Note that I'm not asking  how to use an editing program to achieve this effect after in post production or after the image is captured. facing difficulty to create dynamic xpath for City field of UTest website I am beginner in automation field, I am trying to automated one of the module, I have tried many different XPATH to fetch proper XPATH for City field o UTest site. Im trying to automate registration module as one of my assignment. I have tried my different xpath example below:  //xpath="//body/div[@class='pac-container pac-logo']/div/span[2]/span" //div[@class='pac-item']/span[3] //div[@class='pac-item']/span //div[@class='pac-container pac-logo']/div/span[2]/../span[3]   the following code written:  //xpath to send data in field @FindBy(xpath="//input[@placeholder='Enter a location']") WebElement city;  //xpath for list to appear @FindBy(xpath="div[@class='pac-item']/span") List&lt;WebElement&gt; City;  public void EnterSignupTwo(String CiTy) throws Exception {     city.clear();     city.sendKeys(CiTy);     Thread.sleep(3000);     System.out.println("the size of List " +City.size());//printing list size      for(int i=0; i&lt;City.size(); i++ ) {         String cities = City.get(i).getText(); //storing list element in String         System.out.println("Converted value of city :"+cities); //print of total list      if(cities.contains(CiTy)) //if parameterized match with Available then click     {         System.out.println("enter specific value of city :"+City.get(i).getText());         City.get(i).click();         break;     } }   kindly refer the Which singular names ending in “s” form possessives with only a bare apostrophe? Many questions already ask about this topic ( , , etc.) and their answers vary, but they always give exceptions to the apostrophe-s rule, for example:      6.24 The general rule for the possessive of nouns covers most proper nouns, including most names ending in sibilants."   Examples they give include Kansas’s, Ross’s land, and Jones’s reputation. Exceptions include Jesus’  and Moses’.  Which names does this apply to?  Is the Aeneas’  form correct, or is it Aeneas’s instead? Tense change: previous actions on something that's currently true I'm describing a situation that happened in the past. To explain it, I want to use a description that is both true now and true when the situation happened.  Specifically, I want something like:     She touched me where my neck met my collarbone.   Since my neck is still attached to my collarbone (thankfully), I'm wondering if I shouldn't use the present tense here instead:     She touched me where my neck meets my collarbone.   Which is preferable, and why? The shrine indicator (shrine detector? GPShrine?) has both visual and audio cues, so when I'm near a shrine, it beeps constantly. Unfortunately, I occasionally play the game near other humans who are engaged in their own activities, and my wife is decidedly unimpressed with this feature.   When I got the shrine indicator, I could have sworn that the dialogue said I could press [Y] to disable it... but pressing Y pulls out my weapon.   Ideally, I'd like to just make it stop beeping, but stay on for visual cues. But I'll settle for being able to turn it off entirely at times. Texstudio + Biber + build folder I used  to export my auxiliary files into a different directory. Basically i just add -aux-directory=build in Options -&gt; Configure -&gt; Commands and added the build folder into Options -&gt; Configure -&gt; Build -&gt; Build Options -&gt; Additional Search Paths -&gt; Log. This succesfully generates all the auxilliary files in the build folder and it's also possible to generate the pdf. Unfortunately biber has a problem with this setup:  Prozess gestartet: biber.exe &quot;MyFile&quot; INFO - This is Biber 1.7 INFO - Logfile is 'MyFile.blg' ERROR - Cannot find control file 'MyFile.bcf'! - did you pass the &quot;backend=biber&quot; option to BibLaTeX? INFO - ERRORS: 1  I guess i have to specify the correct path for biber somewhere in Texstudio. Does anyone know where i have to update the path?  someone had similar problems. Also i'm not sure if the solution of   applies to my situation. How do you mock a Sealed class? When I was a kid we had a VHS tape of some Disney live action (not animated) sci-fi movie. It may have been a mini-series or a made-for-tv movie, but I'm not sure. I'm also not sure about the year but it would probably have been late 80s or early 90s based on when I would have seen it. I know it was Disney because they had some live action Goofy and Mickey running around the movie's set as an introduction discussing the plot with a host before the movie.  The plot is basically as follows:   A starship is loaded with mostly children and teenagers as a cross-galaxy human settling effort. Accidents (which are not really accidents and later revealed) cause adult crew members to be killed.  They travel to another planet and face all sorts of troubles. It arrives at the new planet after overcoming obstacles.   Other memorable scenes that may help:   Navigating out of earth's asteroid belt made out of space junk. One of the adults was killed via an airlock malfunction. The ship was supposed to be peaceful with no weapons. At one point they bring on board some crazy guy who puts together a rail gun. There's a memorable line like "Do you want no weapons at all or a well aimed rock?"  There's a planet or space station (I can't remember which) that has a fight scene with the evil bad guy. This scene takes place on what looks to be a carnival or fairground. The bad guys are trying to steal the ship the kids are on because it has some kind of really good futuristic space drive.   Hopefully you remember it better than I do! I've tried looking online for it before with no luck. probability that exactly two accidents happen in any day of a week if 7 accidents happen in a week If 7 accidents happen in a week, what is the probability that exactly 2 accidents happen in any day of the week.  I tried to work it this way.  2 accidents out of 7 can be taken in ${7}\choose{2}$$= 21$.  Now this can happen in any of the 7 days, so we have $7 \times 21$.  Now, the remaining 5 can happen in $6^5$ ways.  So the probability is  $\frac{7\times 21 \times 6^5}{7^7} = \frac{3 \times 6^5}{7^5}$   But this gives me a value greater than 1.   I know I am doing something wrong.  Can anyone help me ? Question on the correlation between two dependent variables I'm working on this question and it's stumping me.     Let $S_n = X_1 + \ldots + X_n$ (with $n&gt;=1$) be a random walk with $X_1, \ldots, X_n$ be iid RV's.   $$ E(X_k)=\mu,\,{\rm Var}(X_k)=\sigma^2.  $$   Find the covariance of $S_n$ and $S_m$   Can anyone help out? I am trying to use the equation: $${\rm Cov}[S_n, S_m] = E[S_nS_m] - E[S_n]E[S_m]$$  but can't quite figure it out.