Respected All  I was trying to prove that  $$\sum_{n\geq 0}(-1)^{n} \frac{1}{2n+1}=\frac{\pi}{4}$$ What I tried to show like this.  We know  $$\frac{1}{1+x^2}=(1+x^2)^{-1}=\sum_{n\geq 0}(-1)^nx^{2n}, |x^2|&lt;1$$ Integrating term by term with in interval of convergence we get  $$\int_0^{x_0}\frac{dx}{1+x^2}=\sum_{n\geq 0}(-1)^n\frac{x_0^{2n+1}}{2n+1}$$ Taking $x_0=1$ we have  $$[\arctan(x)]_0^1=\sum_{n\geq 0}(-1)^n \frac{1}{2n+1}$$ i.e. $$\frac{\pi}{4}=\sum_{n\geq 0}(-1)^n \frac{1}{2n+1}$$  Am I right?
Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?  Can someone point me to a proof, or explain if it's a simple answer?  What I'm looking for is the point where it becomes understood that trigonometric functions and pi can be expressed as series.  A lot of the information I find when looking for that seems to point back to arctan.
Branch of the n-th root of a holomorphic function is holomorphic It is true that given a domain $D$ and a function $f$ that is holomorphic and not vanishing in $D$, then a branch $\varphi$ (with $\varphi$ continuous) of $f^{1/n}$ in $D$ is holomorphic in $D$ ().   My question is, does the statement remain true if we remove the non-vanishing condition? I haven't been able to find a counterexample or a proof on that more general case. SQL Server: Examples of PIVOTing String data Can I be admitted to graduate school in a different field from my degree?  Specific cases include:   If I've taken plenty of advanced courses in field X in the process of completing a degree in another field, can I apply to graduate school in X? What if I haven't taken many courses in X, but I have acquired a good grasp of X through self-study or working in a related field? What if I've never studied X, but I have done very well in an unrelated field?  Could I be admitted to graduate school in X on the basis of general intellectual promise, and then make up the missing background after enrollment?   Note that this question is an attempt to provide a comprehensive answer, to avoid the need for a profusion of field-specific questions on this topic (see the associated ). Please feel free to edit the question or answer to improve them. Given $f_{n} : [a,b]\rightarrow \mathbb R$ is uniformly Lipschitz and $f_{n}\rightarrow f$, how can I show that $f$ is Lipschitz?  $f_{n}$ is uniformly Lipschitz $\leftrightarrow \exists K &gt; 0$ such that for all $n$ and for every $x,y\in [a,b]$ we have $|f_{n}(x)-f_{n}(y)| &lt; K|x-y|$.  $f_{n}\rightarrow f \leftrightarrow \forall~\epsilon&gt;0~~\exists~N$ such that $\forall~n&gt;N$ we have $\mid f_{n}(x) - f(x)\mid&lt;\epsilon~~~(=\mid x-y\mid?)$  Then we have  \begin{align} |f(x)-f(y)| &amp;\leq |f(x)- f_{n}(x)| + |f_{n}(y)-f(y)| + |f_{n}(x)-f_{n}(y)| \\ &amp;\leq |x-y| + |x-y| + K|x-y| \\ &amp;\leq (K + 2) |x - y| \end{align}  which shows that $f$ is Lipschitz. Can anybody tell me if this makes sense? Sorry if layout is poor. Where should I put  tags in HTML markup? When embedding JavaScript in an HTML document, where is the proper place to put the &lt;script&gt; tags and included JavaScript? I seem to recall that you are not supposed to place these in the &lt;head&gt; section, but placing at the beginning of the &lt;body&gt; section is bad, too, since the JavaScript will have to be parsed before the page is rendered completely (or something like that). This seems to leave the end of the &lt;body&gt; section as a logical place for &lt;script&gt; tags.  So, where is the right place to put the &lt;script&gt; tags?  (This question references , in which it was suggested that JavaScript function calls should be moved from &lt;a&gt; tags to &lt;script&gt; tags. I'm specifically using jQuery, but more general answers are also appropriate.) Multiplicative inverses for $Z_n$ 3.8 GPA, but 3 Fs and 1 D on transcript I have a new HP Z800 Workstation, running Windows 7 Enterprise 64 bit. Z800 Specs are .  The machine has 4GB of RAM -- windows reported 1.99GB available, 2GB was "hardware reserved".  I added another 2 GB and windows reports that there is 6GB of physical RAM, but now 4GB is "hardware reserved".  The BIOS sees the full RAM amount as well.  The video card is an nVidia Quadro FX 580, so it shouldn't be using any shared memory at all.    Why is all my extra RAM marked as "hardware reserved" in Windows 7? How to run Windows 10 from an external hard drive? I am currently able to boot from an external drive or multiboot USB. Taking this concept one step further, I have 3 questions:   1) Is it possible to install and run Windows 10 on a portable external hard drive or USB card with the same functionality of being installed on the internal drive?   2) Is it considered legal to do so or will it affect Windows updates in any way?   3) What are the steps to do that? (please outline steps I can follow so I don't lose a drive in the process). :)  EDIT: My question deals specifically with Windows 10, the legality and method of making it a standalone OS running from an external device and what effect if any it has on Windows Update; questions not dealt with in the other question. Can I travel to Canada with valid visa on old/expired passport? Why am I getting "size mismatch" errors from the Iceland mirror? After installing Ubuntu 12.04 (clean install) I fetch a bunch of program via synaptic (installed specially), all programs worked except Gimp 2.8. So I tried purging Gimp, install again (no luck). Checked broken packages and reinstalled them. When using the Software center I get this MSG:  Failed to fetch http://is.archive.ubuntu.com/ubuntu/pool/main/o/openexr/libopenexr6_1.6.1-4.1_amd64.deb Size mismatch   Tried also this:  sudo apt-get purge gimp* sudo add-apt-repository ppa:otto-kesselgulasch/gimp sudo apt-get update sudo apt-get install gimp   And get this MSG:  Failed to fetch http://is.archive.ubuntu.com/ubuntu/pool/main/o/openexr/libopenexr6_1.6.1-4.1_amd64.deb  Size mismatch E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing?   I am wondering if the Icelandic Archive has an outdated version or something?   would really like to install Gimp 2.8 but Gimp 2.6 is OK to.   I am also wondering if I am doing something wrong? I'm trying to run a gunicorn_django process in supervisor but it always exits immediately, giving this error:   INFO exited: my_app (exit status 1; not expected) INFO received SIGCLD indicating a child quit INFO gave up: my_app entered FATAL state, too many start retries too quickly   My server script looks like this:  #!/bin/bash set -e LOGFILE=/var/log/gunicorn/my_app.log LOGDIR=$(dirname $LOGFILE) NUM_WORKERS=3 USER=my-www-user GROUP=my-www-user cd /home/my-www-user/my_app source /home/my-www-user/.virtualenvs/my_app/bin/activate test -d $LOGDIR || mkdir -p $LOGDIR gunicorn_django -w $NUM_WORKERS --debug \  --user=$USER --group=$GROUP \  --log-level=debug --log-file=$LOGFILE 2&gt;&gt;$LOGFILE\  --pythonpath=my_app --settings=settings.active \  my_app.wsgi:application   And my supervisor config looks like this:  [program:my_app] directory=/home/my-www-user/my_app/ user=my-www-user command=/home/my-www-user/my_app/server.sh stdout_logfile=/var/log/supervisor/my_app.log stderr_logfile=/var/log/supervisor/my_app-error.log autostart=true autorestart=true   When I su into the my-www-user account the server starts normally. When I do sudo supervisorctl start my_app it simply hangs until I do CTRL-c, then I find the error above in the supervisord.log file.  Does anyone have a clue about what I am missing? Kali Linux: apt update returns "Hash Sum mismatch" error I'm experiencing exactly the same issue as described in this question: . Before you mark this as a duplicate however, I have tried the solutions posted there, as well as on numerous other sites, including:  sudo apt-get clean sudo rm -rf /var/lib/apt/lists/* sudo apt-get update   Editing /etc/apt/sources.list with alternate official mirrors, such as  deb http://mirrors.ocf.berkeley.edu/kali kali-rolling main non-free contrib or deb https://http.kali.org/kali kali-rolling main non-free contrib  Everything worked after I first imported the VM. I ran sudo apt update and it found some ~650 packages to upgrade. I ran sudo apt upgrade and it encountered an error partway through. That error was solved using sudo apt --fix-broken install, but that is when this hash sum error began. Unfortunately due to hours of troubleshooting I no longer have the details of the earlier error, but I believe it was an error extracting a package due to corrupt data.  I've tried multiple official mirrors, but I get the same error. Additionally, when I downloaded the Packages.gz file  on my Windows machine (VM host) and computed the SHA256 hash, I got the exact hash that apt printed as the expected value. This led me to believe that the error was not with the mirror but with my VM.  The next thing I tried was wget https://mirrors.ocf.berkeley.edu/kali/dists/kali-rolling/main/binary-amd64/Packages.gz followed by sha256sum Packages.gz, which provided yet another different hash output.  To be clear, I have seen 3 different hashes for the same file:   The "correct" hash shown by apt as expected, which is the one that windows also produced after downloading the file using a browser The incorrect hash calculated by apt, which led to the error A different hash calculated by sha256sum after downloading the file using wget using the same URL as for the browser download   I should also note that I have only been referencing the SHA256 hash in each step. The other hash functions are also mismatched when I run sudo apt update, but the file size is the same. I had considered that downloads might be failing due to limited disk space (it is a VM after all) but I don't think that's the case.  What am I missing? What are the most common pitfalls awaiting new users? Covariance of binary and continuous variable Suppose $y$ is a continuous random variable and $d$ is a binary random variable that takes the value $1$ with probability $p$ and $0$ with probability $1-p$.   How do I show that $\text{Cov}(y,d)=(E[y|d=1]-E[y|d=0])p(1-p)$? Sum and product of Martingale processes Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale?  If not, will the two $(X_t)$ and $(Y_t)$ being independent grant their sum and product Martingale? Thanks! What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$? Show that $g\circ f\equiv 0$ implies $\det Df\equiv 0$, where $g(x_1,...,x_n)=x_1^5+...+x_n^5$ Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that $\det Df\equiv 0$.  I was going to start the solution from $n=2$ and then use induction on $n$.  So, for $n=2$, $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a smooth function and $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ where $g(x_1,x_2)=x_1^5+x_2^5$. Then $Df=\begin{pmatrix}   f_{x_1x_1} &amp; f_{x_1x_2}  \\   f_{x_2x_1} &amp; f_{x_2x_2}   \end{pmatrix}$  But now I am stuck in proving the result. Am I in the correct track? Can anyone plese give me a hint to continue?  Ok so since $g\circ f$ is $0$, the image of $f$ is the line $x_1+x_2=0$ which has no min, max or saddle.   If det$Df(x_0)&gt;0$ and $f_{x_1x_1}(x_0)&gt;0$ OR det$Df(x_0)&gt;0$ and $f_{x_1x_1}(x_0)&lt;0$, we have contradiction.   But if det$Df(x_0)&gt;0$ and $f_{x_1x_1}(x_0)=0$, then det$Df(x_0)=0$ or, det$Df(x_0)&lt;0$(Since $f_{x_1x_2}=f_{x_2x_1}$). So this is impossible.  If det$Df(x_0)&lt;0$ we have saddle and it is impossible.   So finally we conclude that det$Df(x)=0$ for all $x$. Am I correct? Order and degree of a differential equation