This is a part of a question to show absolute convergence/divergence of improper integrals. So my professor simplified the question to this $\frac{2}{\pi} \sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ and then proceeded to say that &quot;it is a well known fact that $\sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ diverges&quot;. My question is that shouldn't it converge ? As I can say that the values of the individual terms in the series $\sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ will finally tend to $0$ as $n$ approaches $\infty$. I cannot ask this in class as the lectures are pre recorded and the doubt session is quite far away.
Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.
How can Zeno's dichotomy paradox be disproved using mathematics? A brief description of the paradox taken from :     Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.      This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin.      The paradoxical conclusion then would be that travel over any finite   distance can neither be completed nor begun, and so all motion must be   an illusion.   How can this be disproved using math, as obviously we can all move a walk from one place to another? $\Delta U$, $C_p$, $C_v$ for an ideal gas process Find $f(x)$ for $f(f(x))=\sin x$ Find all (if that's possible) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x))=\sin x.$$  This is a problem a friend gave me. He said he saw it on a book a long time ago, but he couldn't remember what book. I literally have no idea how to solve this so I can't post my attempt here. Im trying to have the bibliography defined for my document using BibTeX and the natbib citation management package. Styles such as unsrtnat give me the references in order of appearance in text. However, I have a problem that I cannot get around. Basically if a reference is supposed to look like:     Tyson J.P. Dikstra M. Holmes F.M.R. Theoretical Analysis on Effect of   Anomalous Low Friction. International journal of Placticity,   34(3-4):413, 1997.   after typesetting it appears as :     Dikstra M. Holmes F.M.R. Tyson J.P. Theoretical Analysis on Effect of   Anomalous Low Friction. International journal of Placticity,   34(3-4):413, 1997.   In other words, it puts the first author in the last place. Is there any method of getting around this?  The entry as it appears in the .bib file (refs.bib) is:  @ARTICLE{Tyson1997, author={Tyson, J.P., Dikstra, M., Holmes, F.M.R.}, title={Theoretical Analysis on Effect of Anomalous Low Friction.}, journal={International journal of Plasticity}, year={1997}, volume={34}, number={3-4}, pages={413}, document_type={Article} }   An excerpt from the tex file:  \usepackage[round,comma]{natbib} \begin{document}  As an alternative method of analysis, \citet{Tyson1997} ...  \bibliographystyle{unsrtnat}  \bibliography{refs}  \end{document} When is it okay to post two answers to one question? I have a lot of 3D objects of various sizes, shapes in blender 2.76b and I would like to set the origin to the bottom center of the objects bounds.  I cannot select the bottom edge and snap to cursor etc as each shapes edges are in different X/Y positions.  Is there a script or tool that will allow me to set the origin to the lowest edge z position? Is there a way to transfer more than one Pokemon at a time? Is there a way to transfer more than one Pokemon at a time to the professor?  Or possibly an addon that gives me this ability?  Vendoring all the trash Pokemon for candies individually is really annoying and time consuming. Adapt BibTeX style to give all initials I'm using a modification of the plainyr BibTeX style. Although my bib-file contains all initials for some authors (e.g. B.D. McKay), I end up with just the first initial in my document (e.g. B. McKay). I thought about editing the BibTeX style myself, but after seeing the code, I decided against it. Maybe there are some people who are more knowledgeable in this area of expertise.  I want my final document to include the initials for all the names, and not just those of the first name.  This is the relevant code of the BibTeX style, or at least, I think it is the relevant code.  FUNCTION {format.names} { 's :=   #1 'nameptr :=   s num.names$ 'numnames :=   numnames 'namesleft :=     { namesleft #0 &gt; }     { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=       nameptr #1 &gt;         { namesleft #1 &gt;             { ", " * t * }             { numnames #2 &gt;                 { "," * }                 'skip$               if$               t "others" =                 { " et~al." * }                 { " and " * t * }               if$             }           if$         }         't       if$       nameptr #1 + 'nameptr :=       namesleft #1 - 'namesleft :=     }   while$ } Trying to identify a short sci-fi story about a human that is "resurrected" by malignant aliens I'm trying to install Linux alongside my 64-bit Windows 8 and went into my partition manager to find this structure.    In order to create partitions for my Linux install I need to free up some space. I see that the last partition is an empty recovery partition, as you can see highlighted, it's 100% free.  I was wondering is it okay to just remove this partition seeing as it doesn't appear I'll be losing anything?  Not sure if this helps but this is on a Fujitsu AH532 running Windows 8 64-bit with UEFI. What does the explicit keyword mean in C++? PDF export on macOS Mojave produces bold fonts biblatex: change name of author according to language? Usually a name stays the same in whatever language. So, whenever I quote a work by Knuth -- be it in an English article or a German one -- I will quote him as Donald E. Knuth.  This is not always the case with ancient or medieval names. They are spelled differently in different languages. Two examples in English | German | French:   Aristotle | Aristoteles | Aristote Thomas Aquinas | Thomas von Aquin | Thomas d'Aquin   This means that when I write an English article, I need a bib-file that contains the English names, but when I want to write a German article, I need to go through my bib file and change the names accordingly. Depending on how often I in fact publish in English or German this could keep me busy and is prone to error.  So, I though, maybe I could have the different names already included in one single bib-file and let biblatex choose the correct name according to the primary language used in the document. Any ideas?  NB: I do not want to use in one single document both spellings Aristotle and Aristoteles (as the following MWE might make you think). Of course, if the article's primary language is English, then all the references to Aristotle will use the English spelling of his name. However, I want to be able to write articles in different languages, without (a) having to edit my bib-files each time or (b) generally maintaing not one single bib-file but two different ones, one for each language, or (c) having two distinct entries in one bib-file  MWE  % !TEX TS-program = lualatexmk \documentclass{scrartcl}  \usepackage{fontspec}         \setmainfont[Ligatures=TeX]{Times New Roman}  \usepackage[ngerman, british]{babel} \usepackage[babel]{csquotes} \usepackage[style=verbose,abbreviate=false,backend=biber]{biblatex}  \usepackage{filecontents}  \begin{filecontents}{\jobname.bib} @book{aristotle:ross:1924,     Author = {Aristotle},    % It is here, I think, that something could be done, e.g., ...     Title = {Metaphysics},     % ... british=Aristotle,ngerman=Aristoteles,french=Aristote     Editor = {William David Ross},     Commentator = {William David Ross},     Introduction = {William David Ross},     Hyphenation = {british},     Location = {Oxford},     Publisher = {Clarendon Press},     Volumes = {2},     Year = {1924}} \end{filecontents}  \addbibresource{\jobname.bib}  \begin{document}  \selectlanguage{british} Some English text.\footcite{aristotle:ross:1924}  \selectlanguage{ngerman} Etwas Text auf Deutsch.\footcite{aristotle:ross:1924}  \end{document} Is it because Rey just saved BB-8 from a net, and then the droid just decides to follow her about - or is it for some other reason?  Does BB-8 have any idea who Rey is prior to being rescued by her?  Just wondering as she literally tells the droid to go elsewhere. Woman planning to visit me from Ukraine to the US applied for a visa and such, claims to have paid for the visa but needs 500 dollars more for the airfare from Ukraine to the US. She says strict rules apply that the airfare cannot be purchased by US person, and must be purchased in Ukraine by person applying for the visa because of terrorist issues, etc..   Is there truth to such? Fastest Gun in the West Problem Integrating Knowledge into an existing website I've noticed that apps (such as Google Contacts Sync) that have been disabled using the stock Android (KitKat) app manager still show up as running when using process observer tools.  This is true even after the device has been rebooted.  Why are disabled apps still running?  Is there an effective (and safe) way to actually disable them?  Solutions requiring root privileges are acceptable.   (Note that for the specific example above, you can tell Android not to sync your contacts, but it still runs the Google Contacts Sync process.  But let's not dwell on that example... it's just an example.) How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space? Stack Exchange app iPhone X optimization The Stack Exchange app doesn’t look very good on Apple’s brand new flagship phone yet.    Also the app can’t identify the device model either. (Says Unknown (iPhone 10,3))   App Version: 1.6.6.2 Device: iPhone X (iPhone10,3) OS Version: Version 11.1 (Build 15B93)