Is $\sum_{k=1}^{\infty}\frac{\cos\sqrt{k}}{k}$ convergent or divergent?
Does $\sum_{n=1}^\infty \frac{\cos{(\sqrt{n})}}{n}$ converge?
Showing that a sequence is monotone and thus convergent Here's the original question:     Let $(a_n)$ be bounded. Assume that $a_{n+1} \ge a_{n} - 2^{-n}$. Show   that $(a_n)$ is convergent.   Okay, I know that if I can show that if the sequence is monotone, I can conclude that it is convergent. But I am not sure how to show that it is monotone.  I know that  $$a_n \le a_{n+1} + \frac{1}{2^n} &lt; a_{n+1} + \frac{1}{n}$$  It looks to me as if it is monotonically increasing but I'm quite not sure how to prove my claim. Any hints would be appreciated. Is there an equivalent in mathematical language to the modulo (or mod) function in computing? Query Activity for ORDER BY xxx DESC LIMIT I'm trying to create a query activity in order to show, from a Data Extension, only the last 5 articles ids. I tried with the formula ORDER BY xxx DESC LIMIT 0,5 as in SQL but it is not working and it is returning this exception:     The ORDER BY clause is invalid in views, inline functions, derived tables, subqueries, and common table expressions, unless TOP or FOR XML is also specified   How can I use ORDER BY in a Data Extension? Coloring specific elements of sets with a prime modified order in an array plot If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is How to fix java.net.SocketException: Broken pipe? Wildcard not working in subprocess call using shlex How to force a node group input field to display a slider? I have created a couple of node groups with some "group-inputs-sliders" (The one called "Roughness" in the bottom picture). I would like it to have the blue-bar slider like the blender native ones. Thanks (: When is the sum of divisors a perfect square? For $n=3$, $\sigma(n)=4$, a perfect square. Calculating further was not yielding positive results.  I was wondering is there a way to find all such an $n$, like some algorithm?  We know that if $n=p_1^{k_1}p_2^{k_2}\dots p_r^{k_r}$ then $\sigma(n)=\frac {p_1^{k_1+1}-1}{p_1-1} \frac{p_2^{k_2+1}-1}{p_2-1}\dots \frac {p_r^{k_r+1}-1}{p_r-1}$. So we have to find $n$ such that $\sigma(n)=\frac {p_1^{k_1+1}-1}{p_1-1} \frac{p_2^{k_2+1}-1}{p_2-1}\dots \frac {p_r^{k_r+1}-1}{p_r-1}=t^2$ for some integer $t$. If each $\frac{p_i^{k_i+1}-1}{p_i-1}$ is a square we will get some such numbers (not all though), but let us consider $\frac{p_i^{k_i+1}-1}{p_i-1}=m^2$.   Working a example, If I take $p_i=17 $ , (or any prime such that $p_i-1$ is a perfect square), then if I can find a $k_i$ such that ${p_i^{k_i+1}-1}=16.m^2$ I will get an $n$, so now when can this happen, is something I have to try out.   But out of curiosity I am asking has this been worked out before. Problem looks broad to me, Is there a solution to this problem? Parse RSS with jQuery I want to use jQuery to parse RSS feeds. Can this be done with the base jQuery library out of the box or will I need to use a plugin? Why does bash say No such file or directory when the file is on the path Restate theorem in separate file I state a theorem in the main body of a document and prove it in the appendix. I use thmtools's restatable environment to restate the theorem there. However, I now need to have my appendix in a separate pdf file (hence a separate tex document). I'm using xr and xcite to get the cross-references and bibliographical references to play nicely.  But the commands created by restatable are not carried. Is there any way to to the restating of the theorem in the appendix in a separate document?  Here's a MWE:  main document: main.tex  \documentclass{article} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  \begin{restatable}{theorem}{thmcmd}\label{thmlabel} Here is the statement of the theorem. \end{restatable}  The above is Theorem~\ref{thmlabel}.  In this way I can restate it: \thmcmd*  \end{document}   appendix: appendix.tex  \documentclass{article} \usepackage{xr} \externaldocument{main} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  I can refer to Theorem~\ref{thmlabel}.  But I cannot restate it like this: % \thmcmd*  \end{document} Where is the operator panel in Blender 2.8? How to format hard disk in Ubuntu 14.04? top x% in [tag] : rules for the tag to be taken into account? Books for particle physics and the Standard Model Grant the OP a binding close as duplicate vote Since the  to the close as duplicate mechanism it is a lot more visible to the OP that there is/might be a duplicate to their question. Empirically, I've noticed a lot more people agreeing, in the comments under their question, that their question is a dupe.   For instance, :        I'm aware that allowing  is  (back in the stone age) but now Stack Exchange is so much larger, and Stack Overflow is groaning under the weight of questions, I think this should be partially revisited.  Based on the principle that the simplest way to keep a site clean is to have as many people as possible nibbling away, I propose that the OP gets granted a binding close-as-duplicate vote.  It should simply be an additional line in the banner, which is . It might be something like this:     This question may already have an answer here:   &nbsp; "" 3 answers   &nbsp; Does this fully answer your question?  Yes / No   I'm not wedded to the Yes / No thing I just can't think of a better way at the moment.  Clicking on Yes would take the OP to the close as duplicate dialogue with that question highlighted. If there are multiple duplicate questions you'd need to ask the OP multiple times. A No could be used to track the relative success of the project, i.e.   Whether the question subsequently gets closed. Whether the question gets reopened again. How many times the OP actually clicks. How many times the OP disagrees with the analysis of the community.   Potentially, a No could be used to open a comment dialogue asking the OP to explain why they think their question is not a duplicate thereby stopping them simply putting "this is not a duplicate" in the question and formalising this process somewhat. Not sure about this idea...  I do not think this should be restricted by reputation, anyone should be able to close their own question as a duplicate. It gives people with lower reputation a more positive experience of the closing process and emphasises the importance of duplicate finding.  Most importantly, self-closes should not, initially, count towards the question ban. We can help new users learn how to use a lot more of the site than they might otherwise see and encourage them to search for duplicates more thoroughly. If the OP then goes on to do the same thing 10 times in a row then you can start using it toward the ban as they're obviously using the users as a personalised search service but at least the first 3-4 times this should not have any effect.  This question is, deliberately, only requesting 20% of the linked question (from 2009) and then only based on the new UI. The linked question asks for binding migration votes etc, which I do not want to see and I am not requesting a binding close-as-duplicate for the OP until someone else has already voted to close (though I think if the OP has 3k rep this might be allowable). This, I hope, addresses the concern's raised in the linked question and explains why this is not a duplicate of that question. How can I check which songs or videos that I purchased from the iTunes Store have DRM restrictions?  All of the mobile OSes are very impressive to me at the moment, and I'm thinking about trying a non-iOS device again. I want to know now the impact it has on my Apple-sourced music collection though.  Is there a date cut-off for purchases when items went DRM free? I need to take a screenshot that records my Eclipse IDE menu item status. For example, if I clicked the "Project" menu and there comes a list of menu items. I need to record those menu items.  The problem is when the menu item list window (which works like a context menu window) has the focus, the screenshot function is not working. I tried everything on this  and nothing works as long as the menu items window is opened. Any mouse click will close the context menu window, so I cannot use mouse either. However if there is no context menu window opened then screenshots works fine.  How can I record the screen with context menu window displayed? How can I install updates and shut down from the command line? I know that there is a simple way to shutdown the computer in command prompt, but is it possible to do system updates and then shutdown the computer in command prompt?   For example, when Vista needs to install updates, it will let me install them and shut down by clicking the Shut Down button (while relegating the normal shut down button into the menu). So is there a command to do this in command prompt?