Does the series $\sum \dfrac{1}{(\ln n)^{\ln n}}$ converges?  I can intuitively say that it converges, because $(\ln n)^{\ln n} $ is going to $\infty$ on a hayabusa
As the title suggests, I'd like to prove that the sum $$ \sum_{n=2}^\infty (\ln n)^{- \ln n} $$ is finite.  The root and ratio test both fail here, but  suggests that there is a comparison that can be used to show convergence.  The only thought I have is that it may help to write the terms as $e^{-\ln(n)\ln(\ln(n))}$, but this has not led me to any particular insight.  Any ideas are appreciated.
How do you round a number to two decimal places in C#? Expanding electromagnetic field Lagrangian in terms of gauge field I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) $3$-cycle is a set of $3$ (distinct) vertices, $a, b, c$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.  I know there are similar ones already on here, but I can't comprehend any of the answers. This is what I have so far:  Base Case: Since the graph has to have at least $3$ vertices to have a simple $3$-cycle, the lowest possible even number of vertices is $n = 4$.   $\dfrac42 + 1 = 3$ degree (edges attached to node). This means each vertex connects to each of the other $3$ vertices since $G$ is a simple graph. Thus, any combination of $3$ vertices will be a simple $3$-cycle.  Inductive Hypothesis: Assume a simple graph with $k$ vertices, where $k$ is even, contains a simple $3$-cycle if each vertex has degree $\dfrac k2 + 1$.  Inductive Step: Since only concerned about when the graph has an even number of vertices, going to show a simple graph with $k + 2$ vertices, where $k$ is even, contains a simple $3$-cycle if every vertex has degree $\dfrac{k+2}{2} + 1$   This is as far as I get and don't know how to finish the Inductive Step (Also please feel free to let me know if I messed anything up, up to this point) Extending my Root Paritition I've Win8/Ubuntu Dual boot. I already have 4 partition so i cannot create a new partition and mount it elsewhere. So, this () question did not help.  Here's a screen shot of my Gparted (run on Ubuntu Live)   How can I extend /dev/sda6  so that  i can make use of unallocated space ?  (w/o failing my current grub) What would you see if you have 2 huge mirrors (you cant look over it or next to it)  You place them with the reflecting side to each other and you look in the mirror from behind one mirror (one way mirror = 1 side glass 1 side mirror, like in a movie when cops are questioning a suspect.)  The mirror would reflect each other endless.  The example could be preformed with a see trough mirror (1 side mirror other side glass) but I don't have one, also I don't have huge mirrors.    Would it make a diffrence if you place a light source and if you dont have a light source?  My guess is that you would always see black because there is nothing to mirror.  Edit important:   There is nothing in the room to reflect on, so it only reflects the mirror, thats what i ment. Is it possible to set the GPU fan speed using the default free ati driver?  My card (X1900) was deprecated in the fglrx drivers a while back.  The free drivers seem to work with 3D, Compiz and such, but running graphics intensive programs makes my GPU heat up and produce artifacts, while the GPU fan never seems to increase. (When set to 100% in Windows it makes a massively louder noise.) Is 45 mins enough in Newark airport terminal C to change the plane for connecting flight? How do I remove a malformed line from my sources.list? I have unistalled and reinstalled the Ubuntu Software Center as per info I found in a similar thread and I got the same response about line 91 or something like that.  I just tried to upload a screen shot but since I'm new it won't allow me to.  I also can not figure out how to cut and paste anything so I have to hand type what the error screen says, both when I attempt to open the software center and nothing happens, when I try to enter commands into the terminal to uninstall, reinstall, whatever I get the same following:  COULD NOT INTITIALIZE THE PACKAGE INFORMATION An unresolvable problem occured while initializing the package information Please report t:his bug against the 'update-manager' package and include the following error message: 'E: Malformed line 91 in source list/etc/apt/sources.list (dist parse)  E: The list of sources could not be read.,  E: The package list of status file could not be parsed or opened.   How do I report bugs?  What can be done about this.  I have searched and everything everyone says to do leads me back to the same line error message.  So, I don't know how to get to line 91 in the source list; to tell you what it says. Sorry, I'm really new to this. That is what I need is to find out how to get there and fix what it says. I would really like to NOT have to re partition my hard drive and start from scratch, so I'm really looking forward to getting this problem solved. I need to be able to install new software. Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+$ Find the sum of the following :   (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$  I tried :  $$\frac{1}{\cos1^\circ}\left[\frac{\cos(1^\circ-0^\circ)}{\cos0^\circ\cos1^\circ} + \frac{\cos(2^\circ-1^\circ)}{\cos1^\circ\cos2^\circ}+...\right]$$  = $$\frac{1}{\cos1^\circ}\left[\frac{\cos1^\circ\cos0^\circ}{\cos0^\circ\cos1^\circ} - \frac{\sin1^\circ \sin0^\circ}{\sin0^\circ\cos1^\circ} + \frac{\cos2^\circ \cos1^\circ}{\cos1^\circ\cos2^\circ} -\frac{\sin2^\circ \sin1^\circ}{\cos1^\circ\cos2^\circ}...\right]$$  For this, as well, I am not getting any pattern to solve further. Please suggest, thanks. TL:DR Smart guides aren't accurate - even though the snap, (end up causing more problems in the long run) is there a way to fix it?  Maybe there are some settings that I'm missing (need to be changed)    I use my smart guides all the time (and am overall pleased with them). I do have my issues with it, as explained here:  A lot of times, when I'm trying to make a bunch of triangles like this:    Done by making a triangle, cloning, flipping, rotating etc. Repeat until full  (Perfect) Snapping using Smart Guides    Ends up looking likebetween the triangles (Outline Mode)    Now I personally (with a strong attention to detail) will notice that there is something off after making just 1 or 2 triangles (Even before outline mode... strokes look oh so slightly thicker/off)   And then I check in outline mode... and sure enough:    Notice the line spaces (and this is in outline mode, so if I didn't notice it before, now it's sitting right in front of my nose)  The way I (seemingly) "fixed" the first image was by adding in a thick stroke outline so that masks out the difference, however, in a lot of my cases I don't want to have a stroke at all (and then you really notice the problems). Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? ASP.NET Call Code Behind Function From Javascript When I type dpkg -l | grep xserver-xorg, I get list of installed packages and their versions:  ii  xserver-xorg               1:7.7+3~deb7u1 ii  xserver-xorg-core          2:1.12.4-6 ii  xserver-xorg-input-evdev   1:2.7.0-1+b1 ii  xserver-xorg-input-kbd     1:1.6.1-1+b1 ii  xserver-xorg-input-mouse   1:1.7.2-3 ii  xserver-xorg-video-radeon  1:6.14.4-8   I am wondering what the number  infront of the colon means, ie the number 2 in 2:1.12.4-6 Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$ I would like to be able to render several files one after the other, when I am not using my computer, e.g. at night. Is there a way of doing this? Is it possible to save a Diablo 3 chat session or save some in game notes I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect? How to theme the maintenance mode page? What would be the best practice for theming the maintenance mode page in Drupal 7? Irreducibility of a polynomial if it has no root (Capelli) Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$.   This  to a related question mentions the result is due to Capelli.  I can prove the result if $F$ has characteristic $p$ as follows.  Suppose $f$ is reducible: $f(X)=g(X)h(X)$ with $g(X)$ an irreducible factor of degree $m$, $1\le m&lt;p$. Then if $\alpha$ is a root of $g$ in some extension field $K$ of $F$, we have $$f(X)=X^p-\alpha^p=(X-\alpha)^p$$ so its divisor $g(X)$ must be of the form $(X-\alpha)^m$. Since the coefficient of $X^{m-1}$ in $g$ is in $F$, we have $m\alpha\in F$.  So $\alpha\in F$ because $m$ is invertible modulo $p$.  How would you show the result in other characteristics? Alternative activity monitors for OS X? I’m a bit intrigued by the new detailed activity monitors and progress indicators in Windows 8 (copying files, the task monitor, network activity, etc.) It got me thinking: surely there must be some better alternatives to the built-in Activity Monitor.app?  “Better” as in more detailed and nicer graphs. Specifically, it would be interested and see percentage disk I/O (out of disk capacity), memory usage, and CPU usage in the same nice way   The Building windows blog has an article  in Windows 8.  I just think that OS X’s task manager — which I have always found superior — just doesn’t cut it anymore compared to what is coming in Windows 8.