I discovered that, for the convergence/divergence of series, the ratio test is stronger than the n-th root test. See  for more details. Since both tests should give rise to the same radius of convergence of any power series, I can deduce that $$ \lim_{n\to \infty} a_n^{1/n}=\lim_{n\to \infty} a_{n+1}/a_n=q $$ if both limits exist. Of course, all $a_n$ are non-negative.  To prove this directly, for all $\epsilon&gt;0$, there exists $N$ such that $q+\epsilon&gt;a_{n+1}/a_n&gt;q-\epsilon$ for all $n\geq N$. Therefore, $$ a_{n+r}&gt;(q-\epsilon)^r a_n, a_{n+r}^{1/r}&gt;(q-\epsilon) a_n^{1/r}. $$ Taking the limit $r\to \infty$ gives $$ \lim_{r\to \infty} a_{n+r}^{1/r}\geq q-\epsilon. $$ Similarly, $$ \lim_{r\to \infty} a_{n+r}^{1/r}\leq q+\epsilon $$ Now let $k=n+r$, so $$ \lim_{k\to\infty} a_k^{1/k}\leq\lim_{k\to\infty} a_k^{\frac{1}{k-n}}\leq \lim_{k\to\infty} a_k^{\frac{1}{k}+\frac{n}{k^2}}=\lim_{k\to\infty} a_k^{\frac{1}{k}}a_k^{\frac{n}{k^2}}=\lim_{k\to\infty} a_k^{1/k}, $$ since $\lim_{k\to\infty} a_k^{\frac{n}{k^2}}=1$. So, $$ \lim_{n\to \infty} a_n^{1/n}=q. $$ The things that are unclear to me is the convergence of there two limits. I can find examples in which $\lim_{n\to \infty} a_n^{1/n}$ exists, but $a_{n+1}/a_n$ ediverges. However, is it possible that $\lim_{n\to \infty} a_n^{1/n}$ diverge when $a_{n+1}/a_n$ converges? Can I modify the above proof to prove that $\lim_{n\to \infty} a_n^{1/n}$ converges when $a_{n+1}/a_n$ converges?
This may have been asked before, however I was unable to find any duplicate. This comes from  of "Mathematical Analysis: An Introduction" by Browder. Problem 14:     If $(a_n)$ is a sequence in $\mathbb R$ and $a_n &gt; 0$ for every $n$. Then show:   $$ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$$   The middle inequality is clear. However I am having a hard time showing the ones on the left and right. (It seems like the approach should be similar for each). This is homework, so it'd be great if someone could give me a hint to get started on at least one of the inequalities.  Thanks.
InDesign: How to restore text-frame position/size to be managed by the master page instead of the actual page? I have some designer that moved slightly a text-frame and in order to be managed again by the master page we need to remove the page and create a new one inserting the content in it.  Is there any different way to do it?  I would like to be able to restore the "power" of positioning/size to the master page because now we need to change the master-page to position the text-frame a little bit to the right and in the case above mentioned will not be updated.  Thanks How can I implement "cut outs" for lighting in OpenGL? So, I'm working with OpenGL (I'm not exactly sure of the version), and I want to do an old-style lighting setup by essentially drawing a black rectangle over the screen, and drawing white circles over the areas that should be lit. I already have the rectangle and the circles, but I can't figure out how to make it so that the white circles are transparent when applied to the black areas. Any ideas? I tried to use blending modes, but it still evades me...  EDIT: Here's the code that I'm using (Python).  view_buf = Buffer(GL_INT, 4) glGetIntegerv(GL_VIEWPORT, view_buf) view = view_buf.to_list() if hasattr(view_buf, "to_list") else view_buf.list  glDisable(GL_DEPTH_TEST)  glDisable(GL_LIGHTING)  glEnable(GL_BLEND)  if logic.lights['smooth']:     glEnable(GL_LINE_SMOOTH)  # # Make sure we're using smooth shading instead of flat # glShadeModel(GL_SMOOTH)  # Setup the matrices  glMatrixMode(GL_PROJECTION) glPushMatrix() glLoadIdentity() gluOrtho2D(0, view[2], 0, view[3]) glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity()  #glAlphaFunc(GL_GREATER, 0.5)  glColor4f(0.0, 0, 0, 0.75)  glBegin(GL_QUADS);                              # Draw Black screen glVertex2d(0.0, 0.0); glVertex2d(view[2], 0.0); glVertex2d(view[2], view[3]); glVertex2d(0.0, view[3]); glEnd()  glColor4f(1, 1, 1, 1)  degtorad = 3.14159/180.0   glBlendFunc(GL_ZERO, GL_SRC_ALPHA)  def DrawCircle(x, y, radius, fill = 1, step = 11):      glVertex2f(x, y)      if fill:         glBegin(GL_TRIANGLE_FAN)     else:         glBegin(GL_LINE_LOOP)      for a in range(0, 360, step):         angle = degtorad * a         glVertex2f(x + (sin(angle) * radius), y + (cos(angle) * radius))      glEnd()   #for layer in range(len(logic.lights['layers'])): #   size =  for light in logic.lights['points']:     DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], step = light[2])     if logic.lights['smooth']:         DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], 0, light[2])  # Reset the state glPopMatrix() glMatrixMode(GL_PROJECTION) glPopMatrix() Whenever I create a document in Microsoft Word, it complains about a lot of my sentences being in passive voice. But, when I read that sentence aloud, it sounds fine to me. I am not sure if it is just me and will a statement in passive voice sound strange to a native speaker?  So, my question is, is it considered bad form to use passive voice generally? Or in some specific cases like written communications only?  Edit: If it is ok to use passive voice, then why does MS-Word complain? Tag badges are re-awarded for some users I'm using the following syntax for my code snippets  \begin{minipage}{\textwidth} \begin{lstlisting}[caption=&lt;something&gt;, label=&lt;something&gt;] The REST Endpoint URL     is http://api.flickr.com/services/rest/  To request the flickr.test.echo service,     invoke like this:    http://api.flickr.com/services/rest/       ?method=flickr.test.echo&amp;name=value  By default, REST requests will send a REST response. \end{lstlisting} \end{minipage}   and I would like to have  \begin{code}[&lt;caption&gt;, &lt;label&gt;] ... code ... \end{code}   which would behave exactly the same.   So I tried this thing but it won't compile and I can't spot the problem  \newenvironment{code}[2][]%     {         \minipage{\textwidth}          \lstset{             basicstyle=\ttfamily,             breaklines=true,             captionpos=b,             frame=bottomline,             extendedchars=true         }         \begin{lstlisting}[caption=#1, label=#2]     }     {         \end{lstlisting}         \endminipage     }   Error when running pdflatex  ! Package inputenc Error: Unicode char \u8:�\expandafter not set up for use with LaTeX.  See the inputenc package documentation for explanation. Type  H &lt;return&gt;  for immediate help. Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$. Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$ for all notnegative value of $x,y,z$.  I think that minimum value is $\frac{3}{4}$ when $x=y=z$ but I have no prove. Recently upgraded to Ubuntu 18.10, having two problems.  Easiest to explain: when I open "settings", go to "privacy", select "screen lock", I get the popup window with the screen lock settings, but everything is grayed out:   Second issue: If I step away from the machine for long enough (seems to be 5 minutes), the screen blanks. Hitting a key or moving the mouse wakes it up. But when it wakes up, the only things on the screen are the icon bar at the left and the top bar with status icons on the right end. The rest of the screen is blue, not my normal lock screen image. The screen isn't locked, it doesn't prompt for a password, it's just...blue.  I can move the mouse pointer around the screen, the icons react if I click on one, I can bring up the popup with volume/brightness controls on the top bar. But if the pointer is on the blue background, neither left nor right button clicks do anything.  Now, if I press ctrl-alt-F1 from there, I get a lock screen where I can enter my password. Then I get the blue screen back, but if I press ctrl-alt-F2 I get my normal screen (with running applications) back.  ctrl-alt-F2 without going through the lock screen first does nothing.  I'm not finding anything while searching that matches either of these sets of symptoms. Suggestions? Prove that $f$ has finite number of roots Let $f:[0,1]\to \mathbb{R}$ be a differentiable function. If there do not exist any $x\in[0,1]$ such that $f(x)=f'(x)=0$, prove that $f$ has only finite number of zeros in $[0,1]$.   I'm not getting any idea how to proceed. I've never really understood why math induction is supposed to work.  You have these 3 steps:   Prove true for base case (n=0 or 1 or whatever) Assume true for n=k.  Call this the induction hypothesis. Prove true for n=k+1, somewhere using the induction hypothesis in your proof.   In my experience the proof is usually algebraic, and you just manipulate the problem until you get the induction hypothesis to appear.  If you can do that and it works out, then you say the proof holds.  Here's one I just worked out,  Show $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^n}{x} = 0$  So you go:   Use L'Hospital's rule. $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0$. Since that's $\displaystyle\lim_{x\to\infty} \frac{1}{x} = 0$. Assume true for $n=k$. $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^k}{x} = 0$. Prove true for $n=k+1$.  You get $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^{k+1}}{x} = 0.$   Use L'Hospital again: $\displaystyle\lim_{x\to\infty} \frac{(k+1)(\ln x)^{k}}{x} = 0$.  Then you see the induction hypothesis appear, and you can say this is equal to $0$.  What I'm not comfortable with is this idea that you can just assume something to be true ($n=k$), then based on that assumption, form a proof for $n=k+1$ case.  I don't see how you can use something you've assumed to be true to prove something else to be true. How is the popularity of "recent tags" calculated? How are these calculated? I see them going up and down all the time, and there are obviously way more than 48 questions about C# so how are these calculated? Static posts page with home.php For my website, I am using a static home page page Home (template front-page.php) and for posts- page News (template home.php)  I need to show all the post titles and exceprts on that page News (templ. home.php), but also show the original title and the_content from that page News itself.  This code below while statement shows all the posts, which is good, but the top the_title also shows the title from the first post, not from the News page itself.  How to achieve what I need?  &lt;?php get_header(); ?&gt;      &lt;?php     the_title();           the_content();     ?&gt;  &lt;?php if(have_posts()) while(have_posts()) : the_post(); ?&gt;          &lt;?php         the_title();         ?&gt;          &lt;?php         the_content();         ?&gt;      &lt;?php     endwhile;     ?&gt;    &lt;?php get_footer(); ?&gt; PS4 Controller not connecting to PS4 and won't reset? I was playing my PS4 two days ago and then packed it up with two controllers that were both connected to this PS4.   I got to where I was going, set everything up and now the controllers won't connect. They show pulsing orange when they plug in, but when I hit the PS Button to connect, it blinks white 1-2 times and goes back to orange.   I have tried resetting them both.  Can anyone help? Can spells or spell-like abilities do non-lethal damage? I'm working on a Paladin/Warlock combination, based off of , and as part of this, I noticed Miniman's remark that Repelling Blast won't be that useful, because as a Paladin, you want people up close and personal when you're in a combat situation, not further away.  However, as a Paladin, your creed sometimes leads you to not want to engage in full-blown combat with a potential adversary -- you may want to drive them off or incapacitate/capture them instead.  With this, I wonder: can Repelling Blast, or other spells and spell-likes (such as invocations), be made to do non-lethal damage?  If so, I'm thinking that it'd be useful in exactly that sort of situation -- halting someone/pushing them away without having to wade into a melee with them or blow them to smithereens. Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?  I remember seeing such a formula, but can not remember where. Any details, (and possibly references) for such a result would be greatly appreciated. I’ve recently changed my ISP and noticed that I no longer can watch video sites like YouTube in real time; I need to wait for buffering first. On both ISPs, my speed was same 5 megabits per second.  I’ve checked normal download of files, and torrents, and they all work fine. Also, checked other video sites, same problem. Used other PC, same problem. Why does (insert favorite search engine here) get better search results than SO, SF, or SU's own search function? I've lost count of the number of times I've either asked a question or been answering a question only to have it closed as an exact duplicate. But searching for the question turns up  a list of seemly unrelated results using the sites builtin search. Then if I switch to a general web search engine and limit its scope to just SO, SF or SU I find the question within the first two or three results.  Why is this?  I'm pretty sure this isn't an exact duplicate of another question though I did find some that were at least somewhat related (using google).  Update:  I also have gotten (somewhat) better results by simply typing a question and browsing the generated list of related questions. Still a search feature that doesn't return good results isn't going to be used by most people.  Common sense would suggest (though common sense isn't always the best indicator) that a targeted solution to a specific problem domain should be superior to a generalized solution. In most cases this appears to hold true. How come Mortarboard is bronze and Enthusiast is silver? :   Silver  Enthusiast    Visited the site each day for 30 consecutive days   56.9k awarded Bronze  Mortarboard   Earned at least 200 reputation in a single day      15.4k awarded    It would seem that earning Enthusiast requires almost no effort while Mortarboard is highly non-trivial. This appears to be reflected in the award count (almost 4:1 in favor of enthusiasts).  So, why is the first one silver and the second one bronze?  (A similar question can be asked about Epic vs Fanatic and probably many other). I'm having trouble understanding the simple voltage regulator that can be built using a zener diode (from section 2.04 in the Art of Electronics).  I know that it would be better to use amplifiers, et cetera, but I'm just trying to understand how this circuit works.      I don't really understand how the circuit works, but I am guessing that when a load is applied to the output, it drains current from the source (Vin) and thus causes the voltage to drop?  How does the zener diode help to maintain the voltage and thus make this circuit act as a regulator? Which prepositions are followed by accusative pronouns instead of reflexive pronouns? I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? How to prove $D^n/S^{n-1}\cong S^n$? In my textbook it is said that the quotient space $D^n/S^{n-1}$ is homeomorphic to $S^n$. I can imagine it for $n=2$, but fail to make a mathematical proof for any dimension.  Can anyone provide a rigorous proof?