I have to show that for any $b &gt;1$, we have  $$ b^n &gt; n$$ for all $n$ sufficiently large, using only very basic analysis (no calculus). My attempt is as follows.    We know that $b^{n+1} - b^n = b^n(b-1)$. For $n$ sufficiently large, say $$n \geq N = \left\lceil \frac{\ln(2/(b-1))}{\ln b} \right\rceil + 1,$$ we have  $$ b^{n+1} - b^n &gt; 2.$$  Now let $\Delta = N - b^N$. Then for any $j\geq 1$, we have  $$ b^{N+j} = (b^{N+j} - b^{N+j-1}) + \ldots + (b^{N+1} - b^N) + b^N &gt; 2j + b^N = 2j + N - \Delta = N+j + (j - \Delta).$$ Thus we have $b^n &gt; n$ for any $n \geq N + |\Delta|$.    This works, but it  seems messy. Is there is better way? I know induction is usual for this type of problem, but establishing the base case for generic $b$ seems difficult.
Prove for any $a&gt;1$ there exist a natural number, $N$, such that $a^n&gt;n$ for all $n \ge N$  My attempt:  By definition, for $N$ to be a natural number $1\le N \le ∞$ for any integer $N$.  By induction, the base case $n=1$ is true because $a^1&gt;1$ which is true.  Now assume that $a^k&gt;k$ is true, then it is also true for $n= k+1$.  Hypothesis  $a^{k+1}&gt;k+1$  Now I am not sure if I am on the right track
How can I minimize Thunderbird 60.2.1 to tray after "MinimizeToTray revived" add-on stopped working? After update, my Thunderbird stopped getting minimized with the "MinimizeToTray revived" add-on. Is there any way to minimize my Thunderbird? Any other add-on that works with the new version? About $\lim \left(1+\frac {x}{n}\right)^n$ I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$ CSS3's attr() doesn't work in major browsers How can I automatically save screen shots by the press on the Prt Scr button? I want a screen shot to be saved automatically to a location I specify, as soon as I press the Prt Scr button on my keyboard, or the Alt+Prt Scr combo. I also want the date and time to be used as file name for each new screen shot. And I want the files to be saved as PNG images.  Is there a Windows program that would allow me to do just this? I don't want any other fancy features like editing and making notes, etc. just a plain simple program that will either automatically save the screen shots to a pre-set location or prompt me where I want to save it. I don't want to go through MS Paint to save the shots.  I guess what I'm after is a little bit of Linux behavior of Print Screen, but only for Windows. And I additionally want the files to be named by the date and time they were taken.  Update:  Meanwhile, I have found one called Greenshot. But I don't know how to get the file name pattern right.  This is the default pattern.  ${capturetime:d"yyyy-MM-dd HH_mm_ss"}-${title}   Can someone explain this? Why do they have the title capturetime and the quotation marks? It produces file names like this.  2013-01-26 15_24_28-Greenshot.png   And this is what it says about the file patterns in the program description.  ${YYYY} year, 4 digits ${MM} month, 2 digits ${DD} day, 2 digits ${hh} hour, 2 digits ${mm} minute, 2 digits ${ss} second, 2 digits ${NUM} incremental number, 6 digits ${title} Window title ${user} Windows user ${domain} Windows domain ${hostname} PC name   How do I get files that include year, month and day, separated by a dash, followed by hour, minute and second? I want to include a time for uniqueness of the file name. Probability that $m$ balls will fall into first box Suppose we have $n$ balls that are randomly distributed into $N$ boxes. Find the probability that $m$ balls will fall into the first box. Assume that all $N^m$ arrangements are equally likely.  Attempt: First, we notice that for the first box, we have $n$ choices, and for the second box we also have $n$ choices, and so on. Thus, we have $n^N$ ways to place the balls into the boxes. Pick $m$ balls out of the total $n$ balls, that gives ${n \choose m}$. In how many ways can these $m$ balls go into the first box? Well, in just ${1 \choose 1 }= 1 $ ways. Thus, $$ P = \frac{ {n \choose m } }{n^N } $$ Is this correct? Why can't I transform faces, edges, or vertices without transforming the whole object? Removing an Ubuntu installation I first had windwos 7, then I installed Ubuntu alongside and when my laptop starts I can choose between ubuntu and windows (the ubuntu boot screen) .  Now I have installed another ubuntu inside the windows 7 using virtual box. I want to remove that first ubuntu installation and free up its partiotion. Can I do it without making any problems for the windows and boot screen? Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$ Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. First, when I never had installed sl:  diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   traslaction: "The program 'sl' is not currently installed. You can install it by typing:"  But when I'm tired of it and I remove it by typing sudo apt-get remove sl per example...  I get this when I type sl:  -bash: /usr/games/sl: Ficheiro ou directoria inexistente   translate: "-bash: /usr/games/sl: No such file or directory" instead of:    diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   This appears with others packages too...  How I get rid of this: -bash: /usr/games/sl: Ficheiro ou directoria inexistente ?  I want see the original:  diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   Output of type sl  sl está hasheado (/usr/games/sl) Question: Can anyone shed some light on this formula? I can't find any information on it.  It has three corollaries that I also need to understand: When we push a box, the box applies an opposite force on my hand, but why does my hand move with the box as I push the box if the net force of my hand is 0? What is happening? Integration of $e^{ax}\cos bx$ and $e^{ax}\sin bx$ Creating compressed GeoTIFF with QGIS Clipper tool? Eclipse doesn't start on Ubuntu 18.04 I installed eclipse via Ubuntu Software. When I start it, it take some seconds and then comes an error window showing the path of the error.  The following is the .log file:  !SESSION Wed May 02 12:53:28 BRT 2018 ------------------------------------------ !ENTRY org.eclipse.equinox.launcher 4 0 2018-05-02 12:53:28.587 !MESSAGE Exception launching the Eclipse Platform: !STACK java.lang.ClassNotFoundException: org.eclipse.core.runtime.adaptor.EclipseStarter     at java.base/java.net.URLClassLoader.findClass(URLClassLoader.java:466)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:566)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:499)     at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:626)     at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)     at org.eclipse.equinox.launcher.Main.run(Main.java:1438)     at org.eclipse.equinox.launcher.Main.main(Main.java:1414)   What is the problem and how can I fix it? New questions, Show Recommended, Sort Newest not working How to make text aligned left/center/right in the same line? I have a laptop with a full keyboard (it has numpad), but the laptop is not that big, and because of that the keys are relatively small, which means I am constantly finding myself pressing keys accidentally on the numpad part.  Is there a way to disable specific keys that I simply don't want? I just finished taking my first year of calculus in college and I passed with an A.  I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on using $u'$), and now that I'm going to be starting calculus 2 and learning newer integration techniques, I feel that understanding the differential in an integral is important. Take this problem for an example: $$ \int 2x (x^2+4)^{100}dx $$ So solving this... $$u = x^2 + 4 \implies du = 2x dx \longleftarrow \text{why $dx$ here?}$$ And so now I'd have: $\displaystyle \int (u)^{100}du$ which is $\displaystyle \frac{u^{101}}{101} + C$ and then I'd just substitute back in my $u$. I'm so confused by all of this.  I know how to do it from practice, but I don't understand what is really happening here.  What happens to the $du$ between rewriting the integral and taking the anti-derivative?  Why is writing the $dx$ so important?  How should I be viewing this in when seeing an integral? Why doesn't my Bash script recognize aliases? How to startForeground() without showing notification?