$\sum\limits_{n = 1}^\infty  {2{a_n}} $ is a series. is it right to say the following?    $$\mathop {\lim }\limits_{n \to \infty } {S_{n = }}\sum\limits_{n = 1}^\infty  {2{a_n}}  = 2\sum\limits_{n = 1}^\infty  {{a_n}} $$
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 $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$?
When cutting/copying some cells to paste them into another sheet/page, sometimes Excel freezes/locks up and fades out. At the top toolbar it says in brackets "not responding".  Eventually, I must click 'X' to close the program. It offers to wait for the program to respond, but never does – it just does nothing until I finally close it, where it offers to recover files etc.   Is there an issue with memory here? What can I do to stop it locking up? Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. I'd like to have Ubuntu installed on a USB stick, that I could plug in (almost) any computer and boot from it, and that I could use (almost) normally, as if it was installed on that computer. I'd also like to have a data partition on it that could be used by (almost) any computer, mostly to store some portable apps, just in case.    I've been told that if I installed Ubuntu in BIOS mode, I wouldn't be able to boot from a computer in UEFI mode, and vice versa.  Since I'd like to be able to boot on any computer, it's a big problem.  What can I do to be able to boot on computers in BIOS or UEFI modes in my case?  I've been searching for solutions but most of them are for specific needs that are different than mine, so it's difficult to extract the minimum requirements from those solutions to apply them to my problem.  Is there a simple, "clean" and quite generic solution to this problem?  I'd appreciate some explainations about the reasons why a specific solution is good or a specific step is needed.    My current setup:   USB stick:  OS: Xubuntu 16.04 i386 installation: full install (from a Live USB built with Rufus) drivers: none proprietary file systems:    partition table:    2: , /boot/efi, 128 Mio, FAT32, [boot,esp] 3: , 128 Mio, HFS+ 4: , 2 Mio, [bios_grub] 5: swap, 512 Mio 6: , /, 24 Gio, ext4, defaults,errors=remount-ro,lazytime,commit=60 1: , /media/data, ~3.6 Gio, FAT32, defaults,lazytime,check=strict  RAM:   /tmp, tmpfs, defaults,nosuid,nodev,noexec,mode=1777   swap:    vm.swappiness=5 swap partition: 512 Mio, for emergency zRAM swap files:    /var/swap/hib.swp: same size as RAM, for hibernation, generated by a script at startup /var/swap/opt.swp: if needed, generated by a script   overprovisioning: ~5.6%   built-in: ~1.2 Gio (apparently) added: 512 Mio  scheduler: deadline Is it okay to have 100% CPU usage constantly? I often do 3D rendering in the background which consumes 100% of the CPU (I lower the priority of the process so that it does not hinder any tasks I am working on).  My concern/question is whether it is okay to run a computer/laptop with a 100% CPU load constantly.  Could this reduce the lifespan of the system? NVMe PCIe x4 SSD on M.2 PCIe x2 slot? i'm about to get a new SSD for my PC, and before spending the money i have some doubts about my decision i would like to clarify First, my mobo is a B450 Aorus Pro Wifi, it has 2 M.2 slots, one of them full length (22110) compatible with PCIe 3.0 x4 and the other one standard length (2280) and support only for PCIe 3.0 x2 About the drive, i'm aiming for a 1tb XPG SX8200 Pro , which is a PCIe 3.0x4 SSD My first M.2 slot is already taken by the main SSD, which i use for the OS and other stuff, i can't really use it. Will a PCIe x4 SSD work on a PCIe x2 slot? How much performance will be lost if it does? Will it be worth it? The blackboard bold font in the AMSFonts package only has capital letters.  I sometimes wish to use a blackboard bold "1", for which I can use \usepackage{bbold}. But this changes the entire blackboard bold font, and I like the original AMSFonts versions of the capital letters better.  Is there a simple way for me to get \mathbb{1} from one package and the blackboard bold capitals from another? Return a blank when cell reference is blank I am using the following formula in a Google Sheets to calculate the number of days between two dates (dates are in columns G and AL):  =ARRAYFORMULA(if(row(A:A)=1, "Readmission Days", (G:G)-(AL:AL)))   It has to be an array because I am working with a Google Forms and entries are automatically populated into new rows.  The problem is that some of the cells in col AL are blank, so the formula is reading those dates as something else and returning a very large number. Also, when both cells are blank, the formula returns a 0, messing up my data.  How can I include something to ignore blank cells or just return a blank when one or both cell references is blank? Do magic item spells require concentration? Do magic items that cast a magic effect such as a ring of invisibility require the user to maintain concentration as if they had cast the spell?  Is this also the case if the item casting the spell is an artifact?   If the item is intelligent can it concentrate on the spell? Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature I am looking for examples of compact manifold of dimension $&gt;2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help? prove that if two real valued  functions are uniformly continuous on some interval $I$  and each is bounded on $I$ then their product $f\cdot g=f(x)\cdot g(x)$ is also uniformly continuous on $I$ . Is boundedness of each function  on $I$ is  necessary for the product Why are my RAW images corrupt but the previews okay? When punctuating the dimensions of something, if both dimensions share the same units, does the punctuation occur after each measurement or just after the last one. For example, if I'm looking at a sheet of paper, which is correct: {8 1/2 x 11"} or is {8 1/2" x 11"}? "Install VMware Tools" greyed out after installing ubuntu 15.04 on vmware player 7.1 "Install VMware Tools" greyed out after installing Ubuntu 15.04 on VMware Player 7.1. I tried checking "No proxy" instead of default "Windows proxy settings" at "File->Preferences->Software updates-> Connection Settings". This did not work either. Is there a way to install VMware Tools? How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero? I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation. Is a URI (specifically an HTTP URL) allowed to contain one or more space characters? If a URL must be encoded, is + just a commonly followed convention, or a legitimate alternative?    In particular, can someone point to an RFC that indicates that a URL with a space must be encoded?  Motivation for question: While beta-testing a web site, I noted that some URLs were constructed with spaces in them. Firefox seemed to do the right thing, which surprised me! But I wanted to be able to point the developers to an RFC so that they would feel the need to fix those URLs. Can't install Linux Headers (Kali Linux) I am trying to install Linux headers for Kali Linux on my machine and I have tried every possible solution on the internet but it always show "Unable to locate packages "  root@kali:/usr/sbin# apt-get install linux-headers-4.6.0-kali1-amd64 Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package linux-headers-4.6.0-kali1-amd64 E: Couldn't find any package by glob 'linux-headers-4.6.0-kali1-amd64' E: Couldn't find any package by regex 'linux-headers-4.6.0-kali1-amd64'   Here is my sources.list file :  # Regular Repositories deb http://http.kali.org/kali sana main non-free contrib deb http://security.kali.org/kali-security sana/updates main contrib non-free # Source repositories deb-src http://http.kali.org/kali sana main non-free contrib deb-src http://security.kali.org/kali-security sana/updates main contrib non-free   Uname -a output :   root@kali:/usr/sbin# uname -a Linux kali 4.6.0-kali1-amd64 #1 SMP Debian 4.6.4-1kali1 (2016-07-21) x86_64 GNU/Linux I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ? "Was" or "were" in subjunctive clauses I'm not a native english speaker, so even though I'm decently proficient at it, I don't really "know the rules" sometimes, and this is one that's been confusing me for a long time.  Which one is correct in each sentence?     If the movement [was/were] to continue uncorrected, the tower would   one day topple.      If I [was/were] rich, I would buy a yacht.   NOTE: I care not only about the case of "I", but also "she", "them", "it", etc, as in the example of the tower. Would it be any different if instead of the tower, it'd be me who'd topple if uncorrected?  I'm pretty sure it's "were" in both cases. That's what they taught me, I think. I started to doubt when I saw a lot of "was", but it sounded like the typical intentional mistake used "stylistically". ("If I was a rich girl...")  Then I saw it some more and thought it came down to an American/British English difference (I was taught British, in theory, and most of what I read is American).    But that tower sentence came straight from "The Guardian"...  When do you use was and when do you use were? How to calculate stays against 90/180 visa rules? Some destinations have a "90/180 rule" which says you can come and go but you must not spend over 90 days out of any 180 days in the area.  concerns this rule.  At least Schengen and Turkey have this rule, probably depending on where the traveller is from and what kind of visa they apply for.  Maybe I'm a dummy but I've always had a lot of trouble trying to add up how many days I've spent in such places when I've entered and left several times over recent months.  Is there an easy way to do this I'm missing or is it difficult for everybody? I'm guessing something based on counting backwards from the end date might be the trick?  Maybe you know of an app or website that can do this for you. "no such file or directory" message when deleting a file that exists on NTFS