Show root test is stronger than ratio test
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
Working with large images and transform in the VSE The keyboard on my laptop seems to be typing by itself:   It sometimes types characters like 134, whether by itself or when I hit certain keys. Repeating characters may appear unexpectedly. The system volume may change by itself, or windows like the Print dialog or Web browser may appear.   This can happen without any input or as I type, and may occur more often when certain keys are pressed. Sometimes, the keyboard fails to respond altogether. This is happening regardless of the software involved.  Is there a ghost in my keyboard—is my keyboard broken?  A sample of text from the keyboard might look like this (normal QWERTY...ASDF...ZXC... sequence is expected):   QWERTYUIOPASDF)(G@*HKLZXCVBN QWERTYIOPASDFGL!#ZXCVB13 QWERTYUIOPASDFGHJKLZXCVB!#N qwertyu41iopasdf19g13h28j113k4113zxcvb4n qwerty4u1iopasdf0g7h1jklzxcvb13m qwert13uuuuuuuuuuuuuuuiopasdf87091134ghhhhhhhhhhhhhhhhhhhhhhh13jk13zxcvb14      This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. Self-Studying Measure Theory and Integration As the title says, I'm interested in reading about measure theory and integration theory on my own time, and was hoping for book recommendations, prerequisites, and things like that.  As for my background, I've taken introductory analysis, where we covered the "typical" subjects up until some theorems about differentiable functions.  I'm taking intermediate real analysis now; we talked about metric spaces in general for a while, but lately we've focused solely on $\mathbb{R}^n$. Now we're working towards a proof of the implicit function theorem in $\mathbb{R}^n$.  I haven't learned any of the theory behind Riemann integration, unfortunately. Is it important to be familiar with this first? How to say the /p/ sound in the middle of words? I'm not sure how to say the /p/ sound in the middle of words like apple, puppy, happy etc. Those "p" sounds have a puff of air or not.  For myself I think those /p/ sounds have a puff of air but less than /p/ in the first of words like pen, pan etc. Is it correct or not?  However, I just listened a word "apple" from a video of Youtube but I can't hear or catch a puff of air from /p/ in apple.  It sounds like /p/ after /s/ like the words spy, spin etc. Set limit to one author when using "et al." in biblatex In the header of my document I set the maximal author names to 2:  \usepackage[style=authoryear-icomp, maxbibnames=9, maxcitenames=2, backend=biber]{biblatex}   Now, when I cite texts with two authors, LaTeX gives me the following:     (Author_A/ Author_B 2012: 232)   When I have three authors or more, LaTeX makes it:     (Author_A/ Author_B et al. 2012: 232)   But I want LaTeX to just name the first author if there are more than two authors ... like this:     (Author_A et al. 2012: 232)   Any tips on how to achieve this ? I created a view to show Users. These users can create nodes of the "Questions" content type.  The view has to render a table to show how many (the total number) Questions the user has created. Something like:  User +++ Questions created John --- 10 Ted  --- 4 Alf  --- 19 Given function $g$ continuous in line $[0,\infty)$, positive and strongly increasing that upholds $\lim_{t\to\infty}  g(t) = L$. Calculate the limit:  $$ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx. $$  At the moment what I think that I should do is swap the $g(nx)$ with $L$ as it approaches infinity but from there on I am just not sure whats my next step is. Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? What did the public at large know about the assassination attempt on Palpatine and how he survived it? If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$   My try:  Note that  $$0+1\times2+2\times2^2+3\times2^3+4\times2^4+5\times2^5+6\times2^6+7\times2^7=1538$$  But the answer is $$n=\sum_{i=0}^72^i=255.$$  How is it derived? What is the correct formula for gravitational time dilation for a satellite in a circular orbit? In the Wikipedia article on  I see two formulae for the gravitational time dilation. One contains $\sqrt{1 - r_0/r}$ but the other $\sqrt{1 - 1.5 r_0/r}$. What formula should I use to calculate the time dilation due to gravity inside a satellite? The speed induced time dilation is of course just special relativity.  With the first of these two formulas I can perfectly reproduce the graphs on time dilation as a function of $r$,  I find that the two contributions to time dilation cancel at 1.5 Earth radii, all numbers are OK... But why not the second formula? Math symbol question: Vertical bar for ''evaluated at ...'' Area of a weird ellipse shape. A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a &gt; b &gt;0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ where $c - d &gt; a + b$ and $d &gt; 0$.  Calculate the total area of the propeller as shown by the shaded region and calculate the average temperature of the propeller, given by $T (r, α) = C/r$, where $C$ is a constant.     Have absolutely no idea how to solve this. Google map only loading half of itself Can I "fill" an empty object? "Congratulation" vs. "congratulations" How to influence the position of float environments like figure and table in LaTeX? I don't know what to do with this. Nothing works. I hope somebody can help me to find a decision $$n!\leq {\left(\frac{n+1}{2}\right)}^{n}$$ What if I accidentally run command "chmod -R" on system directories (/, /etc, ...) I accidently ran  sudo chmod 755 -R /   instead of  sudo chmod 755 -R ./   I stopped it after few seconds, but now there is some problems such as  sudo: must be setuid root   How can I revert permissions back? How to include third party Python libraries in Google App Engine?