Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
I am new to data analysis and now working on a Mixed Effects Logistic Regression Model.  Currently, I have the following data frame (model_data):  Road Id Vehicle_id  entry_time           exit_time          1        1        2017-01-31 00:00:00  2017-01-31 00:00:00 last_veh_time  vehicle_type     status 300            4 wheel          0    Likewise, I have the information (nearly a million rows) of all the vehicles entering a place and exiting the same place.  Status can be either $1$ or $0$ for all of these.  last_veh_time is the last vehicle passed the same place before how much time.    I want to know the probability of a getting status $0$ for a vehicle based on the entry_time and last_veh_time. To do this, I created the model:   model = glmer(status ~last_veh_time + (1 |link ),               data=model_data, REML=FALSE)   Is this the right approach?  Any help is appreciated. I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:  $-y^2dx +x^2dy = 0$  He then asks us to verify that:  $ u(x, y) = \frac{1}{(x-y)^2}$  is an integrating factor.   I multiplied through to get:  $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$   However, the partial derivatives of these do not equal each other so I am a bit confused... Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that $f$ would have a fixed point? What happens in these end-game situations? This is intended to be a canonical question regarding possible endings of the game.  For each question, assume a single proper strike was made, and note where an improper strike would change the game outcome (beyond an additional point for the opponent).  If the Queen has been covered:     What happens if I pocket my last C/m and the Striker? What happens if I pocket my opponent's last C/m? What happens if I pocket my opponent's last C/m and the Striker? What happens if I pocket both players' last C/m? What happens if I pocket both players' last C/m and the Striker?     If the shot covers the Queen:     What happens if I pocket my last C/m and the Queen? What happens if I pocket my last C/m, the Queen and the Striker? What happens if I pocket my opponents' last C/m and the Queen? What happens if I pocket my opponents' last C/m, the Queen and the Striker? What happens if I pocket both players' last C/m and the Queen? What happens if I pocket both players' last C/m, the Queen and the Striker?     If the Queen is left on the board:     What happens if I pocket my last C/m, leaving the Queen? What happens if I pocket my last C/m and the Striker, leaving the Queen? What happens if I pocket my opponent's last C/m, leaving the Queen? What happens if I pocket my opponent's last C/m and the Striker, leaving the Queen? What happens if I pocket both players' last C/m, leaving the Queen? What happens if I pocket both players' last C/m and the Striker, leaving the Queen? I am going to install Ubuntu to dual boot with Windows 8 soon, but one of my friends warned me that partitioning my hard drive will slow my computer down. I run on a 500GB HDD and 8GB of ram if that helps at all. Is that true if I do it? Remove Reviewer and Steward badges Badges should encourage good behaviour. But what when a badge encourages more bad behaviour than good?  We hear it all the time - robo reviewers, badge-hunters. The review system fights with that users that only want badges and not want to help.  There are many suggestions how to find such robo-reviewers or how to make it difficult to robo-review. But why not cut them off at the source? Don't make them want to review for badges at all.  Remove the Reviewer and Steward badges! The Custodian Bronze badge should suffice to bring a user to the review system. And we really only want users doing reviews who want to help and not doing it for the badges. Suppose $k$ is a field, $A$ is a $k$-subalgebra of the polynomial ring $k[x]$. Must $A$ be a finitely generated $k$-algebra?   Thanks. What's a (big) word or phrase to describe a book that is so good that I cannot put down when I start to read it? Why does centering not work with calligra? I'm trying out the calligra package for a wedding order of service. Just trying to make the titles look a little nicer. When I use the \centering command, however, it doesn't seem to work:  \centering Test1\\ \calligra{Test}   Produces calligraphic text that doesn't seem to be centered. Is this fixable?    Thanks all,  James Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. Prevent suggested edits under 10 reputation on protected questions Recently, . The specific  is protected:  protected by  ♦ Apr 27 '11 at 1:10 This question is protected to prevent &quot;thanks!&quot;, &quot;me too!&quot;, or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.  Great, so 10 reputation is required to answer the question. This doesn't stop anonymous and new users from spamming edit suggestions. Could the same restriction be applied to suggested edits to the protected question and its answers? As this would only apply to protected questions, which evidently attract not-so-good attention, it would not hinder new and anonymous users' ability to edit for the most part. Why do you cast downvotes on answers? This is a question which started off with a , then turned into an , and now it's a question in its own right (as suggested by Jeff). I'm not sure where it could go from here :)  It's obvious that different people cast downvotes for different reasons. (Upvotes are less controversial, generally.) Here are some possible reasons:   Answer is wrong Answer is misleading (may be technically accurate, but will lead to readers making mistakes) Answer doesn't match the question (e.g. a C# answer to a question about Java) Answer doesn't really address the question, e.g. suggesting a completely different solution, even if the question specifies that there are good reasons why the current approach has to be followed Answer is okay, but not as good as another one which currently has fewer votes Answer appears to be plagiarised from existing answers. Answer is by someone I don't like. Answer is abusive (profanity, spam etc). No reason, I just felt like it. I'm a , and prefer down to up. Have you considered down? Ideological grounds: answer suggests a technology I don't like, or expresses a negative opinion of a technology I do like   Personally I go with 1, 2 and 8 - although I'll add comments suggesting changes in various other situations.  Have I missed anything? What do you do, and why? What are the net benefits (to you, the world in general, and the answerer) from your approach? Webforms and jQuery, how to match the ID's? I want to use jQuery with asp.net webfoms.  Do I need to get a special toolkit so the .net controls spit out friendly Control ID's?  Reason being, I don't want to write javascript referencing my html ID's like control_123_asdfcontrol_234.  Has this been addressed in version 3.5?  (I remember reading you have to get some special dll that makes ID's friendly). How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? I have played the Witcher 3, exchanged dialogue with 5 trolls already, and decided not to kill them. All 5 trolls were involved in specific quests. I have killed two trolls of which I did not exchange dialogue. I have yet to receive a Troll Mutagen.  How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? Baryons have charges that are the result of a polynomial calculation of their building blocks (quarks)'s fractional charges. But what gives these quarks electric charges? What interactions do they have with photons? And what about leptons, like muons and electrons? How do they get their -1 charge? And then, how do positrons and anti-muons get +1 charge? Same goes for quarks: how does up-antiquarks get -2/3 charge? Or I may want to formulate my question this way:  How does any elementary particle interact with the EM-Force to get an electric charge? Does assigning objects to null in Java impact garbage collection? Bag A has $3$ white and $2$ black marbles. Bag B has $4$ white and $3$ black marbles.  Suppose we draw a marble at random from Bag A and put it in Bag B. After doing this, we draw a marble at random from Bag B, which turns out to be white. Given this information, what is the probability that the marble we moved from Bag A to Bag B is white?  At first since the probability that white is $1/2$ is $2/5$, and the probability that white is $5/8$ is $3/5$, I thought a sufficient answer would be $$\begin{align}\tfrac 1 2 \cdot \tfrac 2 5 + \tfrac 5 8 \cdot \tfrac 3 5 &amp; = \tfrac 1 5 + \tfrac 3 8 \\ &amp; = \tfrac 8 {40} + \tfrac {15}{40} \\ &amp; = \tfrac {23}{40}\end{align}$$, but that was wrong.   Can someone provide me with an answer? Thanks Installing Ubuntu with Fake RAID 1 $z$ is a complex number, by the way.  I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem.  One of the most "satisfactory" things I've done so far is to expand both the real and imaginary parts for the equation \vert z + 1/z \vert^2 = a^2. These parts yield the constraints:  $2xy(1+1/(x^2+y^2)^2)=0$ ;   $x^2-y^2+\frac{2(x^2-y^2)}{x^2+y^2}+\frac{(x^2-y^2)}{(x^2+y^2)^2}=a^2 $  I've graphed the equations and supposedly the intersection should be my set of points, but this method is far from analytical.  What can I do? Any hints on how to approach this? dpkg: error: parsing file '/var/lib/dpkg/updates/0045' near line 0: