The stolz caesaro theorem seems to be a discrete analogue of L'hopitals rule. I can understand l'hopitals rule via the taylor series, that is: $$ \lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{ P(a) + \frac{dP}{dx}|_a (x-a) + O ( (x-a)^2)  }{  Q(a) + \frac{dQ}{dx}|_a (x-a) + O ( (x-a)^2)}$$ Now if $P(a) = Q(a) = 0$, the limit becomes ratio of first order term/ the ratio of values nearby to the point of interest. So, Is there a similar 'series' intuition for
I have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the  and verified that, yes, step B does indeed follow logically from step A, etc. and have internalized the proof to the extent that I can replicate it myself on paper, I still feel that I have made little progress as to why this theorem works the way it does, i.e. what is the intuition behind the theorem, such that it should make sense that it follows the way it does. Therefore I ask: what kind of intuition is there about this theorem? It would also be helpful to understand how the theorem can be used effectively in analysis.  I have seen it referred to as a sort of L'Hopital's rule for sequences, which certainly seems to make sense, but I similarly have little intuitive understanding of how that rule works, either.  I am asking this question not only for my personal understanding, but also for the sake of being able to present it in an illuminating manner, such that the rest of the class can also come away with the same sort of intuition of how the theorem works and how it is useful. Any help would be greatly appreciated.  EDIT: I should point out that the formulation of the theorem we are being tasked to prove is the following:  Let $\lbrace a_n \rbrace$, $\lbrace b_n \rbrace$ be sequences, ${b_n}$ strictly increasing and unbounded. Then if $\lim\limits_{n \to \infty} \frac {a_{n+1} - a_n}{b_{n+1} - b_n} = l$ for some $l \in \mathbb R$, then $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = l$ also.
I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious Making a div vertically scrollable using CSS Any complete group (that is, with trivial center and outer automorphism group) is isomorphic to its automorphism group. The inverse is not true, as the dihedral group of order 8 is isomorphic to its automorphism group but isn't complete.  Problem 15.29 in the Kourovka Notebook asks to find another exemple of a p-group, but I was wondering, is any other exemple known? Is there any other finite (or even infinite) group (not necessarily a p-group) which is isomorphic to its automorphism group but isn't complete? Speed of light travel I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher:      Consider two massless objects moving in the same direction at the speed of light. What would be their relative velocity w.r.t each other?   My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities? Unable to stylize raster data as pseudocolor render type on MAC OSX In a raster layer's Style Properties, I can choose pseudocolor render type, and choose a color ramp and classify the raster.      However, when I click the 'Apply' button, or click 'OK', the visualization of the raster never changes.  I was able to output a 'colorized' version of the same raster using GRASS plugin.  I also reinstalled QGIS 2.8, and was able to adjust and see a raster stylings using the steps I outlined above.  So, can anyone confirm this as an issue, or confirm that there's a mistake somewhere in my process?  I should note that I'm on a Mac, 10.10.4, and I'm using the QGIS provided by , 2.10.1-1 .  Update 1: Here's a screenshot of the rest of my settings.      I have Grayscale set to off in the Style setting.  Also, I don't believe grayscale/color-blind Preview is available on the Mac version of QGIS.  They're does't appear to be the option where it , in the Menu Bar under View / Preview / Photocopy (grayscale) etc.  Either way-- I added a shapefile to show color.  Update 2: I've experimented with setting color interpolation to discrete, linear, and exact, but still am not able stylize using Singleband pseudocolor.  I've also confirmed that the 'Render' box on the the main map view is checked.  Also, I am able to adjust Color Rendering settings, like Hue, Blending Mode, etc., and see the results when clicking 'Apply' or 'OK'.  Update 3: As other Mac users are confirming this, I think it may be a bug. I've created a bug report at the official . Charm Person has verbal and somatic components.  Is there anything in the rules as written that defines how obvious those verbal and somatic components are?  Would it be obvious the caster was casting a spell from 5 feet away?  From across a crowded room?  I'm specifically interested in an answer based on the rules as written, although interpretation and individual opinion are not unwelcome. Returning multiple values from a C++ function Is there a preferred way to return multiple values from a C++ function?  For example, imagine a function that divides two integers and returns both the quotient and the remainder.  One way I commonly see is to use reference parameters:  void divide(int dividend, int divisor, int&amp; quotient, int&amp; remainder);   A variation is to return one value and pass the other through a reference parameter:  int divide(int dividend, int divisor, int&amp; remainder);   Another way would be to declare a struct to contain all of the results and return that:  struct divide_result {     int quotient;     int remainder; };  divide_result divide(int dividend, int divisor);   Is one of these ways generally preferred, or are there other suggestions?  Edit: In the real-world code, there may be more than two results.  They may also be of different types. Derangements with repetitive numbers I have a very simple problem.  Lets assume that I have a well shuffled deck of 52 cards. I start drawing the top card always and when the card matches its rank I lose. J=11 Q=12 K=13.  If there were only 13 cards I could easily use the $\ \frac{!n}{n!}$ for derangements in order to solve this. The problem is that there are 52 cards so when I pass 13 I start from 1 again so I don't know what is the probability to win. Example of the game     1st card: 4 - Continue   2nd Card: A - continue   3rd Card: K - Continue   4th Card: K - Continue   5th Card: 6 - Continue   6th Card: 9 - Continue   7th Card: 10 - Continue   8th Card: A - Continue   9th Card: J - Continue   10th Card: 3 - Continue   11th Card: 2 - Continue   12th Card: 8 - Continue   13th Card: A - Continue   1st card: 5 - Continue   2nd Card 2 LOSE   So actually I have to count from 1 to 13 4 times and if I draw all 52 cards then I win. What's the probability? I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am inquisitive to prove this in a more general way.  The left side of an identity occurs while solving another problem (concerning binomial theorem) so I am more interested in deriving the right side from the left side, else I have to remember it now onward.  EDIT: I am more interested in an algebraic proof rather than combinatorial argument or something involving calculus (however I liked  and  solution), hence I am removing the combinatorics tag. How to change the font color of a disabled TextBox? How can I attach a file to a question? I want to attach a code file to my question, but I am not able to find a link to do it. Is attaching files possible at all? We all know that a simple statement such as:  SELECT * FROM stuff;   should not produce an ordered result. However, when I try to demonstrate this, it always comes out in primary key order.  Moreover a statement such as:  SELECT thing,whatever FROM stuff GROUP BY thing,whatever;   always appears to sort thing by the last field in the GROUP BY clause, which is not at all helpful.  The question is, under what circumstances does SQL SERVER order the result even if unasked, and what can I do to stop this?  I am trying to demonstrate to my students that order is indeterminate unless specified, but this is not helping my case.  I admit that I am working with a small set of sample data.  Thanks Grandi's Series; tends to $1/2$, but why is this considered a valid sum? Compare double in VBA precision problem I have trouble comparing 2 double in Excel VBA  suppose that I have the following code  Dim a as double Dim b as double a = 0.15 b = 0.01   After a few manipulations on b, b is now equal to 0.6  however the imprecision related to the double data type gives me headache because  if a = b then  //this will never trigger end if   Do you know how I can remove the trailing imprecision on the double type? Changing content of footline in beamer How is it possible to set a specific text in the footline of the slides in beamer? In my case, author names and the title are shown. Instead I'd like to show the conference name and country.  \documentclass{beamer} \usetheme{CambridgeUS}  \author{Elena Piper, Amina Shahid, April Fiorese} \title{Evaluating the Change of Performance Metrics for Key ABS}  \begin{document} \maketitle  \end{document} Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$ Number or regions formed when $n$ points on a circle are joined The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$.  This is a fact that I have read in several essays on the dangers of jumping to conclusions in mathematics.  In your opinion, what's the quickest (and/or nicest) way to prove this formula?  Professor  explains somewhere an especially convincing way to tackle this pearl following an idea which a certain high schooler in one of his talks (lectures?) came up with; yet, I don't remember  where it was that I actually read this...  Let me thank you in advance for your insightful replies. "Not being good at swimming, he didn't like it." Could we omit the word "being" here? Not being good at swimming, he didn't like it.   Could we omit the word "being" here?     Not good at swimming, he didn't like it.   Which one is grammatically correct? How to register my unregistered account Possible Duplicate:          I've been using my SO account for couple of days, but I am not registered and I do not want to lose all my questions. How can I register and keep my questions from my unregistered account? Stack Overflow is getting a Meta of its own