Is the series $\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)$ divergent or convergent?
Convergence of $\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets? Is it possible to search for down voted questions? I want to search for new downvoted questions to improve or flag them.   I tried in the searchbar: votes:-1 is:question   but votes:XX only works with positive numbers. I can't query for downvoted questions. Is there a work-around to this? If there's not, should this feature be added? What advantages does using (function(window, document, undefined) { ... })(window, document) confer? How can I reuse existing materials, objects, etc from an existing .blend? Is my baby's gender an example of Schrodinger's cat? If $T^n$ is $q$-contractive, $T$ exactly has one fixed point Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me? 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? I've got a "diagnostics" page in my ASP.NET application which does things like verify the database connection(s), display the current appSettings and ConnectionStrings, etc.  A section of this page displays the Assembly versions of important types used throughout, but I could not figure out how to effectively show the versions of ALL of the loaded assemblies.  What is the most effective way to figure out all currently referenced and/or loaded Assemblies in a .NET application?  Note: I'm not interested in file-based methods, like iterating through *.dll in a particular directory.  I am interested in what the application is actually using right now. How to check if a shell is login/interactive/batch I think I understand the differences between an interactive, a login and a batch shell. See the following links for more help:    (from the sister site: )  (from A User's Guide to the Z-Shell)   My question is, how can I test with a command/condition if I am on an interactive, a login or a batch shell?   I am looking for a command or condition (that returns true or false) and that I could also place on an if statement. For example:  if [[ condition ]]    echo "This is a login shell" fi Can it be proven that : If $a_n&gt;0$ and $\sum a_n$ converges then $na_n \to 0$ (without assuming $a_n$ is decreasing)  Please note This question does not require $a_n$ to be decreasing as condition,  Is it possible to prove $na_n \to 0$ without requiring $a_n$ to be decreasing? I keep thinking that $a_n&gt;0$ and $\sum a_n$ converging then that implies that $a_n$ must be decreasing anyway(not monotonically decreasing necessarily )  Is it possible that $a_n&gt;0$ and $\sum a_n$ converges then $na_n \not \to 0$?  There are already answered questions that assume $a_n$ to be decreasing as a requirement: (this question does not assume $a_n$ is decreasing)       was the post that made me ask this question, a counter example of monotonoic decreasingness of $a_n$ was given in comments by the OP. I see that there a number of questions (, , , , ) on best practices, none of them answer the issue I am going to talk about.  We all need to prepare large documents (or slides) from time-to-time. And once you have reached a certain number of pages, you find that compiling the whole document takes a substantial time, which you need to do to check that the part you have written is error free, specially if it contains a number of images.   To circumvent the situation, I have used the following practices over years to keep compile time at minimum.   Divide the document in to many chapters: The chapters are put in separate files. Only the one I am working one is uncommented.  \documentclass{book}  \begin{document}  %\input {chapterone.tex} \input {chaptertwo.tex} %\input {chapterthree.tex} %\input {chapterfour.tex}  \end{document}    The above technique can also be used for \include.   Keep a preamble replica of the main file in a temporary file (build file): I use this technique when I am preparing a presentation with many slides, say a hundred. Prepare a main file, copy it to another build file. Delete everything between  \begin{document} and \end{document} in the build file. Prepare a slide (or a group of slides) in the build file. Cut and paste the contents between \begin{document} and \end{document} in the build file to relevant location in the main file only when satisfied with the output (image locations, overfull boxes etc.)     We can hear your inputs about the best practices you practice when a you are preparing a document or presentation from scratch.    Now, I have failed to find a good answer in the case when you have finished the whole document or presentation and now you need tweaking the small things, may be edit a text here, add an image there. What will be your best practice?  Perhaps, again keeping only one chapter uncommented will be usable here. But what about large presentations? May be adding a \end{document} just after the point-of-change will be beneficial, but only if you can put that before midpoint of the file.  Commenting a part of the file will also work, but that is error-prone, at least for me. Is the cardinality of the Cartesian product of two equinumerous infinite sets the same as the cardinality of any one of the sets? I couldn't find this explicitly stated in any handout or text. This certainly seems to be true from the examples I have seen:  The Cartesian product of two infinitely countable sets is again infinitely countable. The Cartesian product of two sets with cardinality of continuum again has cardinality of continuum. Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework:     Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable.   From what I understood, denumerable means that it is infinitively countable. There are some  on the web about this topic, but I am still not understanding their explanation, maybe because they are not explained so easily (for me).  From what I understood, a set is denumerable if there's a relation with the natural numbers (but I am still not understanding what is this relation).  I have heard also about equinumerous sets (contain a bijective function = onto + one-to-one function), but I cannot relate this information with the problem to try to solve it. Iterating over Java collections in Scala I'm writing some Scala code which uses the  API. I would like to iterate over the rows contained in the java.util.Iterator that I get from the Sheet class. I would like to use the iterator in a for each style loop, so I have been trying to convert it to a native Scala collection but will no luck.  I have looked at the Scala wrapper classes/traits, but I can not see how to use them correctly. How do I iterate over a Java collection in Scala without using the verbose while(hasNext()) getNext() style of loop?  Here's the code I wrote based on the correct answer:  class IteratorWrapper[A](iter:java.util.Iterator[A]) {     def foreach(f: A =&gt; Unit): Unit = {         while(iter.hasNext){           f(iter.next)         }     } }  object SpreadsheetParser extends Application {     implicit def iteratorToWrapper[T](iter:java.util.Iterator[T]):IteratorWrapper[T] = new IteratorWrapper[T](iter)      override def main(args:Array[String]):Unit =     {         val ios = new FileInputStream("assets/data.xls")         val workbook = new HSSFWorkbook(ios)         var sheet = workbook.getSheetAt(0)         var rows = sheet.rowIterator()          for (val row &lt;- rows){             println(row)         }     } } How do I change the email address used for Google Play? My email address I use for the Play Store will be invalid soon, so I need to change it. However I don't want to make a new account because I want to keep all my apps. How can I do that? Exterior power "commutes" with direct sum I know that for vector spaces $V, W$ over a field $K$, we have the following identity :  $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ which holds in a canonical way : for $0 \le k \le n$, the maps  $$ \Lambda^k(V) \times \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W), \\ (v_1 \wedge\cdots \wedge v_k,w_{k+1} \wedge \cdots \wedge w_n) \mapsto (v_1 \wedge \cdots \wedge v_k) \wedge (w_{k+1} \wedge \cdots \wedge w_n) $$ factors through the tensor product's universal properties, which gives us a map $\varphi_k : \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W)$, and since the category of $K$-modules has biproducts, we get a canonical map  $$ \varphi = \bigoplus_{k=0}^n \varphi_k : \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \longrightarrow \Lambda^n(V \oplus W) $$  What if I replace $K$ by a (commutative ring with $1$) ring $A$? Does this map have a kernel? I know it is surjective (it suffices to "look at it", i.e generators of the codomain are obviously all hit by $\varphi$). I don't know if it helps, but I am working in integral domains. In the field case, this is obvious if $V$ and $W$ are finitely generated, since one can use dimension arguments. I wouldn't mind if the above identity held only in the finitely generated case either. If it worked in general it would be cool though.   Note that when trying to compute the inverse map with the universal property, there seems to be a problem... my guess would have been $$ (v_1 + w_1,\cdots,v_n + w_n) \mapsto \bigoplus_{k=0}^n (v_1 \wedge \cdots \wedge v_k) \otimes (w_{k+1} \wedge \cdots \wedge w_n) $$ and this map is obviously $A$-multilinear, but there is a problem with the alternating property ; I get some terms left. I'm guessing this is the wrong map... if there is one. How can i check if $remote_addr IP is NOT in CIDR range in nginx? I am using nginx latest web server. How can i check if the $remote_addr ip is NOT in specified CIDR range and return appropriate error code.  i would like to return specific error codes if IP is NOT in specific CIDR IP ranges.  how can i do this for CIDR IP range ?  e.g.    If remote_addr is not in 123.123.123.123/24 then return 444 code . I am trying to understand how to see the total size of my hard drive &amp; how much is left. It was recommended on various searches to use the command below   df -h Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.6M  1.6G   1% /run /dev/sda1       219G   29G  180G  14% / tmpfs           7.8G   18M  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup cgmfs           100K     0  100K   0% /run/cgmanager/fs tmpfs           1.6G   64K  1.6G   1% /run/user/1000   However I don't understand any of these directories. I mean what are they, are these all directories or partitions or what? And why do they appear under "mounted on"? How does that bit work. I assume the /dev/sda1 is my solid state drive? So now if I install applications, which one of these directories does my software get installed  into? I assume it just gets installs in the / directory (I haven't even got a clue what the / directory is). Can someone explain what the other directories are and what they are used for? I own a 2016 Macbook (with the USB-C charging) - at the time and in response to all of the reports of poor cables and poor chargers, I bought a meter so I could see what the throughput of my cables/chargers was. However, having run some tests I'm now not sure what to consider normal / what is harmless to the battery.   For all tests I used an original Apple C-C cable - though was able to replicate with two third party cables I own.  I have the original Apple block charging at 14.3 V / 1.99 A and a third party block that charges the same.  Another third party block which advertises USB PD and 3 A is currently charging at 20 V / 1.43 A  I've done some research online but I can't find any articles that indicate what a normal USB PD output is relative to the devices that connect to it and would appreciate any guidance / assistance.   TL;DR: Have Apple charger which charges at one voltage/amperage. Bought several others advertised at PD which are higher voltage, lower amperage. Not sure what is 'safe', or 'fast', or does least overall damage to battery. I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?