How to calculate $$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt[n]{{{3^n} + {4^n}}}} \right)?$$    I saw on the suggestions that you can use the sandwich theorem or compression. And the fact that $\displaystyle\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt[n]{2}} \right) = 1$  So I must first find upper bounds and lower.  someone help me with this please?
How can I prove the convergence of the sequence $b_n=\sqrt[n]{x^n+y^n}$ where $x, y &gt; 0$?  Can I divide it in two cases?  Case 1: $x &gt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{x^n+x^n} = \sqrt[n]{2 \cdot x^n}=x \cdot \sqrt[n]{2} $$  Case 2: $x &lt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{y^n+y^n} = \sqrt[n]{2 \cdot y^n}=y \cdot \sqrt[n]{2}$$  Result: Does the sequence converges to $\max(x,y)$?
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$ I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files? Why would Mystique's DNA allow for Sentinels that can duplicate mutant powers? Implement a Selective Comment Ban Statistical interpretation of Maximum Entropy Distribution I have used the principle of maximum entropy to justify the use of several distributions in various settings; however, I have yet to be able to formulate a statistical, as opposed to information-theoretic, interpretation of maximum entropy. In other words, what does maximizing the entropy imply about the statistical properties of the distribution?   Has anyone run across or perhaps discovered yourself a statistical interpretation of max. entropy distributions that does not appeal to information, but only to probabilistic concepts?  As an example of such an interpretation (not necessarily true): "For an interval of arbitrary length L on the domain of the RV (assuming its 1-d continuous for simplicity), the maximum probability that can be contained in this interval is minimized by the maximum entropy distribution."  So, you see there is no talk about "informativeness" or other more philosophical ideas, just probabilistic implications. How to disable all caches in safari 11? I am using Safari to develop for a while. One thing I always liked is when you clear the cache, it really does, unlike other browsers I tried in the past. But in previous versions, specially when working in a code in a local testing server, just reloading the page will reload everything from disk. Now I have to use de shortcut command+alt+e to clear before reload or some new CSS won't load, even some http request will return old values. Just reloading the page won't guarantee to show me the last modifications.  I looked at preferences and develop menu but didn't find an option to just don't use any caches at all - the perfect option will be don't use caches when I have the web inspector opened, means I am developing.  Did I miss this option, or there is another way to disable all caches? USB stick mysteriously become write protected So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. MWE:  \documentclass{minimal} \usepackage{commath, tikz} \begin{document} \begin{tikzpicture} \node {$:$}; \end{tikzpicture} \end{document}   When I compile this, pdfTeX enters an infinite loop. Why? Usage and origin of "sister" in expressions like "sister company, sister ship, sister site" etc How to create LV(Logical Volume) after ubuntu is installed? everyone. I need to create LV for xen guest. There is only one disk of 70G. While installing, I use the default process and not the LVM. Do I need to reinstall Ubuntu to create the LV? Is there some better method? Thanks a lot.  root@vswitch-BlackfordESB2:/home/vswitch# vgdisplay   No volume groups found root@vswitch-BlackfordESB2:/home/vswitch# vgcreate vm /dev/sda1   No physical volume label read from /dev/sda1   Can't open /dev/sda1 exclusively.  Mounted filesystem?   Unable to add physical volume '/dev/sda1' to volume group 'vm'. root@vswitch-BlackfordESB2:/home/vswitch# pvdisplay  root@vswitch-BlackfordESB2:/home/vswitch# df -h Filesystem            Size  Used Avail Use% Mounted on /dev/sda1              60G  4.2G   53G   8% / udev                  3.6G  4.0K  3.6G   1% /dev tmpfs                 1.5G  1.1M  1.5G   1% /run none                  5.0M     0  5.0M   0% /run/lock none                  3.6G     0  3.6G   0% /run/shm Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know how to verify this statement. $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$ Let $p$ be an odd prime, and let $g$ be a primitive root modulo $p$. Prove that $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$.  I've seen this solution on the math stack exchange but most of the solutions use quadratic reciprocity. Unfortunately, I skipped that section because it seemed very hard, but the proof that  $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$ seems easy with it.  Is there another way to solve this This question is proposed in the chapter of Finite Abelian Groups.  DEF:In elementary number theory, an integer $g$ is called a primitive root for the modulus $n$ if $\mathbb{Z}_n^\times$ is a cyclic group and $[g]_n$, is a generator for $\mathbb{Z}_n^\times$. I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. Differences between LuaTeX, ConTeXt and XeTeX Use feature as module? Does anyone have experience with using a feature as a module? The usage I am talking about is strictly placing hooks inside the .module file. Can that trigger any sort of bug? If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. Allow linking to specific revision of revision history showing diff Currently, the only way to link to a revision (clicking "link"), you just get  which shows what the post looked like at that revision:        Personally, I rarely use this and haven't ever found myself in a scenario (as far as I remember) where I would prefer the way it currently works to what I'm suggesting here.  Could we:   Add another link, or Change the current link   Such that it:   Displays the diff on a separate page (I'd prefer this, since the highlight and scroll thing doesn't always work properly - although perhaps it has something to do with loading images, so it might not apply here, or, if it doesn't, we could fix that instead), or          Highlights and scrolls to a diff, like what is done when you link to an answer () Why are there two ticker symbols for Google — GOOGL and GOOG?  They seem to be mostly identical, but their market values are different: currently, 551.64 for  and 548.79 for . Calculate business days I need a method for adding "business days" in PHP. For example, Friday 12/5 + 3 business days = Wednesday 12/10.  At a minimum I need the code to understand weekends, but ideally it should account for US federal holidays as well. I'm sure I could come up with a solution by brute force if necessary, but I'm hoping there's a more elegant approach out there. Anyone?  Thanks. I got the following problem: Right now I am checking my equations, which are usually written in align. This looks great in most of the cases, but sometimes the pagebreaks are weird, ie there is a lot of vertical whitespace left (like 2.5cm) at the end of a page. I made the following MWE to illustrate this.  \documentclass[a4paper, 12pt, headsepline]{scrreprt} \usepackage[onehalfspacing]{setspace} \usepackage{amsmath, amssymb, mathtools} \usepackage{mathptmx} \usepackage[a4paper]{geometry} \geometry{left=2cm,right=5cm,top=2cm,bottom=2cm} % \usepackage{lipsum} %\lipsum \usepackage[pangram]{blindtext}  \begin{document} \Blindtext[1][3]%1 paragraph, 3 pangrams \begin{align} X_{t+1} &amp;=  \frac{X_{t+1}+X_{t+1}}{Y_t}  \label{eq:1}  \intertext{Solving...:} \notag  X_{t}  &amp;=  \frac{X_{t+1}+X_{t+1}}{Y_{t+1}} \notag \\ \intertext{Solving further...:} \notag  % ===PAGEBREAK HERE: 2nd page begins=== &amp;= \frac{X_{t+1}+X_{t+1}}{Y_t} +  \frac{X_{t+1}+X_{t+1}}{Y_t}  \notag \\ &amp;= \frac{X_{t+1}+X_{t+1}}{Y_t}...  \notag \\ \intertext{Solving further...:} &amp; = \frac{X_{t+1}+X_{t+1}}{Y_t}  \notag \end{align}  \end{document}   In the MWE, the first 'Solving further' should be on the first page instead (to save space and it would also look better I guess?). The position where I would like the pagebreak to be is also marked as a comment in the MWE.  My question is first on a general basis: Which is the best environment to achieve multiple line equations, which can be aligned as in align, and to save space at the same time? Or to be more precise: If possible, I'd like to stick with align and adapt the pagebreaks only if necessary.  Thanks!