If $\sum _{n=1}^{\infty \:}a_{n\:}$ converges and $\sum _{n=1}^{\infty \:}b_{n\:}$ converges.  how to proof that $\sum _{n=1}^{\infty \:}a_{n\:}-b_n$ also converges?
Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$?
I'm on a new computer and I haven't downloaded Data Loader in years but now, I'm having all kinds if issues.  So apparently, I have to download Zulu JDK 11 (msi) first, Then download and unzip Data Loaded and install. Which opens a command prompt... After I follow all of the command prompt steps, then I get an error.  Am I supposed to install Java too? And if so, do I download Java or Zulu first?  Apparently a lot of people are having this same problem with no resolve.  Any help would be much appreciated. What happens to matter in a standard model with zero Higgs VEV? Why don't my female dwellers give birth to children? I am playing Fallout Shelter on my IPad. There were some women who have already gave birth, but two female dwellers are carrying their babies for about 18(!) hours. I know that this process takes three hours, so this is not normal. What do I do? Grub menu at boot time... "holding shift" not working I read here at  that holding Shift during boot can bring up the GRUB menu. However I try holding one or the other Shift keys, tapping, tapping then holding, nothing works. Am I missing something? How does one access the GRUB menu during boot time?   No dual boot here, just standard Lubuntu 14.04 install. How do I disable tabs for  tag I am using &lt;a&gt; tags for links on a web page. How do I disable the Tab key from selecting either of them? In Section 9 of Chapter 6, Lang constructrs for the first time a non-abelian extension as a splitting field $K$ of $X^n - a$ when the ground field $k$ does not contain primitive $n$-th roots of unity $\zeta_n$. He constructs an injective homomorphism of the Galois group $G(K/k)$ into the matrices of type   $$ M=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix} \quad \hbox{with}\ b\in \mathbb{Z}/n\mathbb{Z} \quad \hbox{and} \quad d\in (\mathbb{Z}/n\mathbb{Z})^*,$$  where $(\mathbb{Z}/n\mathbb{Z})^*$ the multiplicative group of integers modulo n.  Afterwards, he states in Theorem 9.4 that if $n$ is an odd positive integer prime to the characterstic and $[k(\zeta_n):k]=\varphi(n)$ (i.e, the primitive roots of unity are not in the ground field), then the above homomorphism is, in fact, an isomorphism. Besides, the commutator group is $G(K/k(\zeta_n))$, so $k(\zeta_n)$ is the maximal abelian subextension of $K$.  Lang proves the theorem inductively over the number of prime factors in the decomposition of $n$ (i.e. if $n=9= 3\cdot 3$, then the proof "would take two steps" to complete). The induction start with $n$ prime is written clearly and I agree with it.   Now come the questions:   I do not understand Lang's argument to prove that the commutator subgroup is the subgroup $\begin{pmatrix}1 &amp; 0 \\ b &amp; 1\end{pmatrix}$, $b\in \mathbb{Z}/n\mathbb{Z}$. What is this projection on the diagonal and how does one see that the factor group of the commutator group is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$?   By the way, direct computation of $x\, y\, x^{-1}\, y^{-1}$ for two arbitrary matrices $x=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix}$, $y=\begin{pmatrix} 1 &amp; 0 \\ c &amp; e \end{pmatrix}$ yields a matrix $\begin{pmatrix} 1 &amp; 0 \\ b(1-e)+c(d-1) &amp; 1 \end{pmatrix}$. As $n$ is odd, $2 \in (\mathbb{Z}/n\mathbb{Z})^*$, so taking $e=1$ and $d=2$, we indeed reconstruct the above-mentioned group.   The second (inductive) part of the proof is a complete mystery to me with virtually every sentence being a Theorem which needs an extra proof. Can someone give me some hints?   2.a. Lang writes $n=p\,m$ and takes $\alpha$ to be the root of $X^n-a$. It is clear that $\beta=\alpha^p$ is a root of $X^m -a$. Now comes the argument "By induction we can apply the theorem to $X^m-a$". I agree that we can apply it here but the result is that $K=k(\beta,\zeta_m)$ is the splitting field of $X^m-a$. In the diagram below, there appears $k(\beta,\zeta_n)$ out of nowhere.   2.b. Now Lang proceeds to apply the theorem on $X^p-\beta$ over $k(\beta)$. Bergman's "Companion" reminds to check that $[k(\beta,\zeta_p):k(\beta)]=\varphi(p)$ before applying the theorem and gives a hint "$[k(\zeta_p):k]$ and $[k(\beta):k]$ are relatively prime". I disagree with this hint. Take $n=21= m p$ where $m=3$ and $p=7$. Then $[k(\zeta_p):k]=\varphi(p)=p-1=6$ but $[k(\beta):k]=m=3$ and both are not co-prime. Can we apply the theorem here at all? Or if so what is the argument? If Bergman's argument is valid, where is the fault in my logic?  2.c. Even assuming that we can apply the theorem in 2.b., then we can say something about $k(\alpha, \zeta_p)$ and not $k(\alpha,\zeta_n)$. How do we proceed to checking everything Lang states in the last section of the proof (basically, everything after the diagram)?  EDIT: I have found the essentially same question Is there any easy way on Windows to log %CPU time per process over time to a file for later analysis?  As far as I can see...   Task manager shows me the %CPU per process but only visually - no way to save to a file. Perfmon will let me save to a file, and will additionally create a time-based file (taking snapshots at specified time intervals) - which is almost what I want, except that as far as I can see it can't break down the %CPU by process. Process Explorer will let me break down by process, but it will only save the file as a snapshot of a single point in time   So, none of these are quite what I need: I need to get a file that contains something like 'top 10 processes by CPU, every 15 seconds, until I stop the monitoring' The reason I need this because I have a machine on which some process is causing occasional brief spikes in CPU usage several times a day and I need to find out which process is the culprit.    Can anything do that, or have I missed some feature of perfmon or process explorer? So I was taught that:  Kinetic Energy (of electron) = Energy (of photon) - Ionization Energy  If E(photon) = IE, then KE=0 of the electron.   What does this physically/theoretically mean?   My current thoughts/interpretation is that enough energy/force is applied to ionize the electron so it is 'sufficiently far' from the atom, and then I guess it just moves with whatever speed it is moving at with natural kinetic laws, since no more energy/force is being applied....?  Any clarifications would be greatly appreciated, thanks in advance. Commuting: How to carry office clothing so it doesn't get crushed? I have the bike, I have the best route; but now I want to know, how do I transport my clothing in each day? I typically wear suit pants and a business shirt, but not a suit jacket.   Am I better off leaving my clothes at work, or is there a special bag that would let me carry in shirt/pants without them getting crushed? After I factory reset my Nokia 6, this icon appeared. I've never seen it before. What is this icon in the highlighted part? How do I remove the file suffix and path portion from a path string in Bash? Given a string file path such as /foo/fizzbuzz.bar, how would I use bash to extract just the fizzbuzz portion of said string? Drawing a constant randomly distorted circle I am trying to draw a special and contradictive plot : A circle with random distortions which have to remain constant regardless of how many times I build the document.  An example of what I am trying to achieve can be seen below    This figure was produced with the following code  \documentclass{standalone} \usepackage{tikz}    \usetikzlibrary{decorations.pathmorphing}   \begin{document}     \begin{tikzpicture}         % axes         \draw[ultra thin] (-2.5, 0) -- (2.5, 0);         \draw[ultra thin] (0, -2.5) -- (0, 2.5);         % undistorted sphere         \draw[ultra thin, dashed] (0, 0) circle (2.25cm);         \draw[ultra thin, dashed, -&gt;] (0:0) -- (125:2.25);         \node at (-1, 1) {$R$};         % distrorted sphere         \draw[very thick] plot[domain=0:350,smooth cycle ] (\x:2+rnd*0.5);         \draw[very thick, -&gt;] (0:0) -- (45:2.1);         \draw[ultra thin, -&gt;] (1, 0) arc (0:45:1);         \node at (1.2, 0.4) {$\theta$};         \node at (0.7, 1.25) {$R(\theta)$};     \end{tikzpicture} \end{document}   The problem with the code is that whenever I make a change to it and rebuild the document, the "distorted" circle will have a different shape, since I am using the rnd function. The reason I would like to have a constant-random shape is because I would like to denote its radius at an angle θ, therefore it has to touch the outline of the circle.  Any idea on how to achieve such a thing? This movie came out around 2000 and was created for one of the cable networks (and never in the theaters).  I think it was on the original Sci-Fi Channel or maybe the USA Network, but really not sure.    It's about a 1950's American-style utopian world.  The main character, if I remember correctly, worked at a car dealership.  I recall him looking like Jon Hamm, Adrian Pasdar, or Scott Bakula, but I may be way off there (and I didn't see the movie in their filmographies).  He drives his nice car home, and the whole town is beautiful, as are its people and white picket fence style homes.     He suddenly starts seeing weird things.  For example, instead of people driving cars, they're actually riding rickety bicycles.  The manicured, paved streets are actually dirt roads.  Houses are, in reality, tiny shacks.  Instead of people's clothes being immaculately maintained, everyone's really wearing filthy rags.  And instead of delicious food, they're eating something that looks like rotten garbage.  Something is making everyone believe they live in a beautiful, perfect society, but it's mind manipulation that's almost on the scale of the movie "The Matrix."    He befriends a pretty lady, and together they travel outside their town and try to find answers.  They even meet an outside militarized government, whose officials are interested in understanding and helping him.  This other city's infrastructure appears to be state-of-the-art and well funded.  The officials all have nice uniforms.  However, their environment is also under the same influence that's making everyone see things differently than what they really are.  The main character somehow sees through this illusion too, and realizes this fake prosperity is affecting others outside his home town--maybe the entire world!    The movie ends with       the man and woman eventually running into some elderly maintenance man in a forested area.  The maintenance man's job is to service a machine that alters everyone's perception of reality.     I posted about this a couple of years ago on another site, and somebody took a guess that it was "The Prisoner" but unfortunately that's not it.  I didn't really explain it well the first time, so hopefully this is clearer. Unable to install rJava in R 3.0 in Ubuntu 13.04 Can regular expressions be used to match nested patterns? Why do we say that the Earth moves around the Sun? In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.  But then in physics we learn that movement is relative, and it depends on the reference point that we choose.  Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?  Was movement considered absolute in physics back then? Why did the texture I painted in texture paint mode disappear? With Ubuntu One shut down, is there a way to sync applications between computers? Since Ubuntu one is no longer it appears that Sync between computers no longer works.  Is there a new method to do this?  I'd like to sync the apps from one machine to another. Material setings in properties window is missing Distance Between Two Points in Matlab