Product formula for $\sum_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n}{k}$
Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
Recursively replace string in files I want to replace the backslash in the string page_path\example_filename.txt with a forward slash.  I also want to be able to run this on a large file system and have it recursively search all directories.  I found someone on the web who said to use grep, xargs, and sed but I wasn't able to get it to work. I've been trying different variations on delimiters and escape characters, but didn't get it.  Here is the command I was trying to run: grep -lr -e 'page_path\\' * | xargs sed -i 's/page_path\/page_path//g'  Thanks in advance. What's interesting in latus rectum? I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks! Number of automorphisms of a direct product of two cyclic $p$-groups What can I do to manage my phone's internal storage? If you're not used to seeing the 'low on internal storage' notification, what is it that you're doing to keep your phone's internal memory from filling up so quickly?   The 'SD Card and Phone Storage' menu in the settings doesn't really tell me much other than how much is free. And, I don't really have that many applications installed. The top 5 storage hogging apps on my phone are Gmail (18.6 MB), Twitter (17.5 MB), Facebook (12.55 MB), Maps (9.63 MB), and Swype (6.06 MB).  So, are there apps out there (besides your normal file manager ones, like Astro) that will help monitor internal phone storage? Is there something I should be doing to keep my phone storage use to a minimum? Show that $S + T$ and $aT$ are linear Let  $S:  V \to W$ and $T: V \to W$ be linear transformations. Given $a$ in $\mathbb{R}$, define functions $(S+T): V \to W$ and $(aT): V \to W$ by $(S+T)(v)=S(v) + T(v)$ and $(aT)(v)= aT(v)$ for all $v$ in $V$. Show that $S+T$ and $aT$ are linear transformations. The Next tag badge block overlaps the badge score block. Cannot get specific WMS to display in Leaflet or OpenLayers 3 Alternate property name while deserializing Can I use my Windows 7 license in a dual-boot configuration? Possible Duplicate:          I've purchased a retail license of Windows 7 Ultimate for my machine at home.  I'm thinking of creating a dual-boot install:   64-bit install for work apps (need the RAM for virtual machines and RAMDisk) 32-bit install for entertainment apps (Media Centre, games, etc)   How does retail licensing work in such situations? Can I activate the same license on both installations?  Edit:  I would like to add that I want to have separate partitions for work and play. Show that $2^n > n^2$ whenever n>4. Let $S(n)$ be the proposition "$2^n &gt; n^2$"  Basis step: $S(5)$ is true, since $2^5 = 32$, $ 32 &gt; 25 = 5^ 2$  Inductive step:  Induction hypothesis:   Assume $S(K)$ is true, $2^K &gt; K^2$.  Then  $2^{K+1} = 2.2^K &gt; K^2 + K^2$   I don't know how to proceed after this. How to complete this proof? How to disable gedit side panel? How do i disable this stupid side panel of gedit, that somehow appeared? There is nothing in the settings or plugins. Tried rm -r -f ~/.config/gedit as well, with no success. I don't have big screen and i find it totally useless and eating up what little free space for editing documents i have. See Screenshot I am wondering if my proof is correct? Thank you for whoever willing to take a look at it for me.     Proof $\liminf E_k \subset \limsup E_k $   If $\{E_k\}_{k=1}^\infty$ is a sequence of sets, we define \begin{align*} \limsup E_k &amp; = \bigcap_{j=1}^\infty\left(\bigcup_{k=j}^\infty E_k\right)\\ &amp;= \bigcap_{j=1}^\infty(E_j \cup E_{j+1} \cup \cdots)\\ &amp;= (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots. \end{align*}  Therefore, $\limsup E_k$ consists of those points in $\mathbb{R}^n$ which belong to infinitely many $E_k$. \begin{align*} \liminf E_k &amp; = \bigcup_{j=1}^\infty\left(\bigcap_{k=j}^\infty E_k\right)\\ &amp;= \bigcup_{j=1}^\infty(E_j \cap E_{j+1} \cap \cdots)\\ &amp;= (E_1 \cap E_2 \cap \cdots) \cup (E_2 \cap E_3 \cap \cdots) \cup \cdots. \end{align*} Therefore, $\liminf E_k$ consists of those points in $\mathbb{R}^n$ which belong to all $E_k$ for $k \geq k_0$.  Now consider $x \in \liminf E_k$, then $x \in E_1 \cap E_2 \cap \cdots$, or $x \in E_2 \cap E_3 \cap \cdots$ and so on, that is to say, $x \in E_{k_0} \cup E_{k_0+1} \cup \dots$ for some $k_0$. More specifically, $x \in E_k \forall k \geq k_0$. Therefore, $x \in (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots$, so we showed $x \in \limsup E_k$. What is the advantage of using wp_mail? What is the advantage of using wp_mail() over mail(). Codex says they're similar, but they seem to be very similar. I can clearly see that $\dfrac{n!}{n^n}\to 0$ when $n\to\infty$. But how do I know if the sum $$\sum_{n=1}^{\infty}\frac{n!}{n^n}$$ is convergent or not? I know this might be basic, but thank you if anyone can help me. Align multiple numbered aligned blocks There are many discussions over how to align multiple align environments with interspersed texts. However, I want to label multiple aligned blocks such that each number will center vertically for each aligned blocks and at the same time align across blocks.  Here is a MWE that achieves either at one time. The first part simply uses one giant align environment with some \nonumber directive. The second one use equation to wrap aligned but lose alignment info. How should I achieve the best of both worlds?  \documentclass{article} \usepackage{amsmath} \begin{document}   % Part I   \begin{align}           x^2 + y^2 &amp;= 1 \\                 z^2 &amp;= 0 \nonumber \\     x^2 + y^2 + z^2 &amp;= 1   \end{align}    % Part II   \begin{equation}     \begin{aligned}       x^2 + y^2 &amp;= 1 \\             z^2 &amp;= 0 \\     \end{aligned}   \end{equation}   \begin{equation}     \begin{aligned}       x^2 + y^2 + z^2 &amp;= 1     \end{aligned}   \end{equation} \end{document} How to draw arrows between circled coefficients of polynomials using tikz Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? Why can't Python's raw string literals end with a single backslash? Mirrored object not symmetrical I am trying to mirror an object and the original is symmetrical along all axis, but when I attempt to mirror across the y axis, the mirrored side is tilted to a different angle. Difference between staticmethod and classmethod What is the difference between a function decorated with  and one decorated with ?