Beamer document jumping of alignment
Avoiding jumping frames in beamer
Line of Bones to Travel and Conform to Curve (Train or snake following path) I've searched google extensively on this. Can't seem to find an answer.   I know about spline IK, which is wonderful for conforming a row of bones to a path. But combining the armature object with a follow path modifier causes the point of origin to move, but the bones themselves stick to the beginning of curve.   Ultimately my goal is to make a creature like a snake or centipede follow a curve path, thus animating it fairly realistically ... How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$? I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.  The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:   Substitute $y = x - 1$. This leads nowhere, I think. Find the Taylor polynomial. Makes no sense, it is a polynomial. Divide by major term. Dividing by $x$ got me nowhere. Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$. Simplify the expression. I do not see how I could. Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.   If somebody sees a simple way, please do let me know.    Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:   $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.   It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.  This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :). Please give me feedback on my answer to this question. Question: For all $ n\geq1:\binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n} $   is equal to $ \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1} $. Answer: False, since;  Let $a=1,  b=-1$  and $n=1$  ,$k=0$   , using Binomial expression.  $$(a+b)^n=  \sum   \binom{n}{k}  a^{k}b^{n-k}$$  Then $(1+-1)^{2(1)}=\binom{2}{0}1^{0}(-1)^{2-0}+\binom{2}{1}1^{1}(-1)^{2-1}+\binom{2}{2}1^{2}(-1)^{2-2}=\binom{2}{0}-\binom{2}{1}+\binom{2}{2}$  Thus :For all $n\geq1:  \binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n}   \ne \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1}$  . Re-Structuring a JSON Add table row in jQuery How did humans and vulcans communicate in Star Trek: First Contact? I've heard Google's algorithm is based on PageRank, and that it has something to do with the links to your site. Can someone explain it, and more importantly, how I can increase my site's PageRank? Getting Started with Unit Testing Unit testing is, roughly speaking, testing bits of your code in isolation with test code. The immediate advantages that come to mind are:  Running the tests becomes automate-able and repeatable You can test at a much more granular level than point-and-click testing via a GUI    My question is, what are the current &quot;best practices&quot; in terms of tools as well as when and where to use unit testing as part of your daily coding? Lets try to be somewhat language agnostic and cover all the bases. "sudo -u postgres psql postgres" does not ask for password How can I set an external display (monitor) as the primary one? Currently the MacBook's display works as primary, so every application I open gets open on the 13" and I have to drag it to the 24". I also don't have access to the dock on my 24".  Is is possible to turn off the MacBook screen and use only the one attached via DVI?  (I've just connected my 24" HP LP2475w via Mini DisplayPort to DVI to my MacBook Pro 13", which works great except for this one thing.) cron job schedule Is it ever bad to use \left and \right? For mathematics, is it ever disadvantageous to use \left(  and \right)  in equations?  I can't see any way that doing this would hinder the typesetting, so why isn't () just converted into \left( \right) automatically? This would save a lot of typing.  Cheers! Set Snap Mode with PyQGIS Each  has a compiled cost that contributes towards the  compiled character formula limit.  What are the current compiled costs associated with each formula function? Visual Studio - How to change the return value of a method in the debugger? When I'm debugging, I often have to deal with methods that does not use an intermediate variable to store the return value :     private int myMethod_1()    {       return 12;    }     private int myMethod_2()    {       return someCall( someValue );    }   Well, in order to recreate a bug, I often have to change values on the fly. Here, I could want to see what happens when myMethode_1 return zero. Same thing for myMethod_2.  Is there a way to do that without modifying the code ? What I would like to do, it's to place a breakpoint on the return line (or on the closing bracket) and type-in a new return value.  Note : I'm using Visual Studio 2005 and Visual Studio 2008.  Thank ! I'm looking for an example of a finitely generated module with submodules that are not finitely generated.  I've found  dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.  And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.  Can you please help? Thank you! I read a novel when I was 12 or 13 (late 80's, early 90's) about two timelines combining on Earth.  It was a full length novel for adults that I swiped from my mom (she doesn't remember it at all.)  I've been trying to figure out what the title is for some time but my google-fu is unable to find the right search terms.    The book was published in the 80's or perhaps early 90's.  The story: It was set on Earth and featured two timelines running concurrently (in the 80's?)  Timeline 1:  The nuclear missile crisis had gone badly and a low level nuclear war had more or less destroyed the US, but small towns still survived but are slowly fading away.  Much of the story was about the scarcity of goods, and the protagonist in the main thread of the book started the story by finding a 6 pack of Coke buried in a ruined house.  Throughout the story people were finding things that had grown scarce (deer, bullets, human beings.)  One strange scene I recall involved mutant/radioactive sweet potatoes that had driven a man insane.  Eventually people in this timeline began having dreams of their lives in Timeline 2.  Some of the people were not having dreams, and it was eventually discovered that they had died in Timeline 2.    Timeline 2:  There is panic in a world recognizable as our 1980's as nuclear war draws near.  Since more and stronger missiles have been developed, the end of humanity is at hand.  Things start disappearing from that world as the war begins.  Timeline 2 was mostly seen as vignettes of panicked people running around.  Combination of Timelines: In a weird, incongruous, almost surely poorly written scene, Gaia (yes, Earth itself) is seen to be combining a dying timeline with a doomed timeline in order to save humanity.  At the end of the novel, kids from Timeline 2 are transported into Timeline 1 in order to repopulate.  The new kids drink all the Coke, in case you were wondering...  Any help finding the title would be welcome.  I recall enjoying the book a lot when I was a kid, but I suspect my mom traded it in at our local used book store because at some point it just disappeared. How is an empty set truly "empty"? In a  question, an answerer says:  an empty bag is a bag with nothing inside it.  Makes sense, but I'm reading a textbook right now that says:  The empty set has only one subset (namely, itself).  Which seems odd to me. If a set is empty, why would it contain a subset? Regardless of whether or not the subset is also an empty set, the &quot;empty set&quot; still contains a subset, so how is it really &quot;empty&quot;? To go back to the empty bag analogy, it just sounds like my text book is saying:  an empty bag is a bag with nothing but an empty bag inside it.  Which doesn't make sense because then the bag isn't empty. I can't wrap my head around this concept. Operator precedence in a find command? See below example:  $ ls -1p appledir/ applefile orangedir/ orangefile $ find . -type f -name apple\* ./applefile $ find . -type f -name orange\* ./orangefile $ find . -type f -name apple\* -o -name orange\* ./applefile ./orangedir ./orangefile $ find . -type f \( -name apple\* -o -name orange\* \) ./applefile ./orangefile $    I was surprised to discover I needed the parentheses for this to work as expected; apparently I haven't internalized the rule of precedence by which find evaluates its arguments.  How can I easily predict when I will and when I will not need to use parentheses to explicitly group find primaries?  Put another way, what are the rules by which I can imagine find inserting parentheses into the commands I give it, which will allow me to accurately predict how it will evaluate ungrouped expressions? In what order do multiple readied actions with the same trigger resolve?