Why does the $\lim \sum{n^{(1/n)}-1}$ diverge as $n\rightarrow \infty$? I suspected it would converge as $\lim {n^{(1/n)}}=1$ as $n\rightarrow \infty$ but computation show otherwise. So, now I am trying to show the partial sums are unbounded but didnt get far.
Let $a_n=\sqrt[n]n-1$. Does $\sum_{n=1}^\infty a_n$ converge?
Chat reputation in left bar seems different than any other reputation displayed for users what is the relationship between the dynamical casimir effect and virtual particles? Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$ i have to prove:   $$\begin{vmatrix} -2a &amp;a+b  &amp;a+c \\   b+a&amp;-2b  &amp;b+c \\   c+a&amp;c+b  &amp;-2c  \end{vmatrix} = 4(a+b)(b+c)(c+a)$$   I have tried many calculations between the rows and columns of the determinant to get to the answer i want to solve the exercise however none of them gave me something right. Can someone help? INACTIVE_OWNER_OR_USER Let $a&gt;0$, show that for $x&gt;0$, $1&lt;f(x)&lt;2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$  I could take the derivative, find the maximum of the function and conclude. But that involves heady algebra and you have to solve a high degree polynomial. Is there an easier way simply using inequalities? Thanks, much appreciated Ubuntu Software doesn't open in Ubuntu 20.04 If i click on Ubuntu Software, it doesn't open. I have a fresh install of Ubuntu 20.04. I tried try to upgrade all packages, but it doesn't fix this problem.  If I try to run killall software-center there is no process with this name, and software-center returns command not found. If I try to reinstall it with apt I have these results:  $ sudo apt-get --purge --reinstall install software-center Reading package lists... Done Building dependency tree        Reading state information... Done Package oracle-java8-installer is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source E: Package 'software-center' has no installation candidate dpkg error: "trying to overwrite file, which is also in..." Is there a difference between single and double quotes in Java? Is there a difference between single and double quotes in Java? How to invoke vim editor and pipe output to bash Show that $\langle x,y\rangle_A = \langle Ax,Ay\rangle$ is an inner product on $\mathbb R^n$ Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb R^n$. Then  $\langle x,y\rangle_A$ is an inner product iff  1)  $\ker A = \{0\}$.  2) $\operatorname{rank} A= n$.  3) all eigenvalues of $A$ are positive.  4) all eigen value of $A$ are non negative.  Clearly if $\ker A \neq \{0\}$, then there exist $x \neq 0$ such that $Ax =0$, then $\langle Ax, Ax\rangle_A = 0$, then $\langle ,\rangle_A$ is not inner product, so $\ker A = \{0\}$, thus rank of $A$ is $n$.  I am unable to check the 3) and 4) option. Please help to check the other option. Thank you Create Case thread Id using Apex (SOQL limit-proof) There are many posts about creating Case threadIds with Apex, notably   As the posters have noted, the SFDC rules of the threadId are subject to change over time and the only reliable way that I've seen is to do   // To be replaced with a real case Id. Id caseId = '500c0000008kJL5';   List&lt;Messaging.RenderEmailTemplateBodyResult&gt; renderResults =   Messaging.renderEmailTemplate(null, caseId, new List&lt;String&gt;{'{!Case.Thread_Id}'});  System.debug('this is the Case Thread Id: ' + renderResults[0].getMergedBody());   HOWEVER ... The method call Messaging.renderEmailTemplate burns a SOQL ()  So you can't prepopulate your Cases in an afterInsert or beforeUpdate Trigger because a transaction can be up to 2,000 (Platform Events), up to 10,000 (direct Apex DML) or more, typically, 200 rows.  All of these are > 100 SOQL limit.     So, before I go to the trouble of doing this in a chained set of async   tasks; is there more recent thinking on this subject of creating ThreadIds? Exit strategies for "chameleon questions" Normalising Constant for exponentiated function What would the normalising  constant be of the following, or atleast an approximation? I would like to avoid sampling.  $$f(\theta)=\exp(-k_1e^{-k_2\theta^2}-\theta^2)\qquad\theta\in(-\infty,\infty), \quad k_1,k_2&gt;0$$  In the event that its not possible, what would be a valid polynomial lower bound on $\log(f(\theta))$, that would take into account $k_1$  For anyone, who is interested, I end up with this expresssion because I place a Gaussian prior over (a squared exponential) kernel parameters, and trying to use variational bayes (hence, a lower bound). I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics  in general.  What are some freely available great physics books on the Internet? edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.  As a starter:         Table of contents sorted by field (in alphabetical order): OpenGL GLX context is not using direct rendering, which may cause performance problems. (steam) AMD DRIVER When I start steam in a terminal, I get the following error messages   charlie@SteamMPL:~$ steam rm: no se puede borrar «/home/charlie/.steam/steam»: Es un directorio rm: no se puede borrar «/home/charlie/.steam/bin»: Es un directorio Running Steam on ubuntu 14.04 64-bit STEAM_RUNTIME is enabled automatically [2014-08-03 11:50:13] Startup - updater built Jul 15 2014 18:14:26 Installing breakpad exception handler for appid(steam)/version(1405474565_client) libGL error: failed to load driver: swrast Looks like steam didn't shutdown cleanly, scheduling immediate update check Installing breakpad exception handler for appid(steam)/version(1405474565_client) [2014-08-03 11:50:14] Checking for update on startup [2014-08-03 11:50:14] Buscando actualizaciones disponibles... Installing breakpad exception handler for appid(steam)/version(1405474565_client) [2014-08-03 11:50:14] Download skipped: /client/steam_client_ubuntu12 version 1405474565, installed version 1405474565 [2014-08-03 11:50:14] Nothing to do [2014-08-03 11:50:14] Verificando instalación... [2014-08-03 11:50:14] Performing checksum verification of executable files [2014-08-03 11:50:15] Verification complete Gtk-Message: Failed to load module "overlay-scrollbar" Gtk-Message: Failed to load module "unity-gtk-module" Installing breakpad exception handler for appid(steam)/version(1405474565_client) Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element Fontconfig warning: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 78: saw unknown, expected number [0803/115015:WARNING:proxy_service.cc(958)] PAC support disabled because there is no system implementation libGL error: failed to load driver: swrast Error: OpenGL GLX context is not using direct rendering, which may cause performance problems.   Hardware information:   CPU: AMD Athlon x4 760k  GPU: AMD Radeon R7 250 1 GB GDDR5 (Driver: Catalyst 14.4) RAM: 8 GB DDR3 1600   What can I do to circumvent this issue? Possible Duplicate:             I want to add some shell scripts in my launcher as shortcuts..  They are located deep inside and i have to navigate to it every time to execute them...  I tried  link where the 2nd answers seems to solvee it but as said in comments neither can i find any option to add a launcher..  I guess it existed in 11.04 with a "+" button on launcher but i use 12.04.  So how do i get this working?? And i need those commands everytime for sure..! How to create a custom UI? I was watching this  on making a custom UI tab using Python, and in the line from bpy.types import Menu, Panel, UIList, I see the names of objects that look like they can all be used to make different UI elements. Does bpy.types contain all the objects that people use when they want to make custom UI elements? How to load image to android and resize Multiple "order by" in LINQ Finding the remainder of $\overbrace{11\ldots1}^{123 \text{ times}}$ divided by $271$