I have to find the following sum, and I've solved up to a certain point but cannot proceed further. Any hints on how to proceed would be helpful.  $$\sum_{r=0}^{n}\frac{(-1)^r}{(r+1)^2}{n\choose r}$$   Consider the expansions of $(1-x)^{n}$ and $(1+x)^{n}$. $$\begin{align}\int_{0}^{x}(1-x)^{n}\mathrm dx&amp;=\sum_{r=0}^{n}\frac{(-1)^{r}x^r}{r+1}{n\choose r}\\\frac{1-(1-x)^{n+1}}{n+1}&amp;=\sum_{r=0}^{n}\frac{(-1)^{r}x^r}{r+1}{n\choose r}\tag1\end{align}$$ Similarly integrating $(1+x)^{n}$ from $0$ to $x$. $$\frac{(1+x)^{n+1}-1}{n+1}=\sum_{r=0}^{n}\frac{x^r}{r+1}{n\choose r}\tag2$$ Replacing $x\mapsto 1/x$ in $(2)$, the required summation is just the coefficient of $x^{0}$ in the product of $(1)$ and $(2)$, which simplifies to the following. $$\text{coeff. of }x^{n+1} \text{ in } \left[\underbrace{(1+x^{n+1}-x^{n+1})}_{=0}-(1-x^2)^{n+1}+\underbrace{x^{n+1}(1-x)^{n+1}}_{=1}\right]$$ Any hints on how to proceed further. Thanks.
How can I calculate the following sum involving binomial terms:  $$\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$  Where the value of n can get very big (thus calculating the binomial coefficients is not feasible).  Is there a closed form for this summation?
Print lines between (and including) two patterns Snap application doesn't see files from another partition I have installed KeePassXC snap package on Ubuntu 17.10. I have multiple linux systems installed on the same drive, so I have made most of my home folders like ~/Documents created as links to a separate ext4 partition (users have same uid so permissions are correct). I tried to open my passwords' database which I have inside Documents, but keepassxc cannot see any files inside Documents. If I put the file in my home folder then it sees it.  Are snap packages able to find files outside from my home folder? Any way to make KeePassXC (and the rest snap packages) access the files at my separate ext4 partition (maybe some configuration somewhere)? List of Figures + Beamer? I want to have a List of Figures in my Beamer presentation.  I spent some time searching the web for a solution and only found desperate people telling others that it's not possible, i.e.  or .  I also made some experiments on my own, i.e. "abuse" tocloft to do the job. However, even \usepackage{tocloft} causes pdflatex to fail with ! LaTeX Error: \l@section undefined. (and lying to tocloft by defining the command as empty beforehand doesn't to any good either).  I can't really see why it should be that difficult to make a List of Figures in beamer, since \tableofcontents works flawlessly.   I'm not afraid of doing some coding on my own, but I'm not deep enough into TeX to really understand the magic behind the "List of Figures", "Table of Contents" and alike.  Is anyone out there having suggestions or links that might help me figure this out?  \documentclass{beamer}  % uncomment the following line % to get the error message % `! LaTeX Error: \l@section undefined.`  %\usepackage{tocloft}   % uncomment the following lines INSTEAD % to lie to tocloft % and get the next error message  %\makeatletter %\newcommand\l@section{} %\makeatother %\usepackage{tocloft}  \begin{document}  \begin{frame}{A frame} \begin{figure} \mbox{A} \caption{An 'A'} \end{figure} \end{frame}  \begin{frame}{List of Figures} \listoffigures \end{frame}  \end{document}   EDIT: Solution  Starting from Marco Daniels answer to my question, I modified his code to reflect my needs. Here is what I came up with. Please feel free to comment, improve and reuse this.  \RequirePackage{ifthen}  \makeatletter  \AtEndDocument{%   \clearpage   \beamer@tempcount=\c@page\advance\beamer@tempcount by -1%   \if@filesw   \newwrite\tf@lof   \immediate\openout\tf@lof\jobname.lof\relax   \newwrite\tf@lot   \immediate\openout\tf@lot\jobname.lot\relax   \fi }  \long\def\beamer@makecaption#1#2#3#4{%   \def\insertcaptionname{\csname#1name\endcsname}%   \def\insertcaptionnumber{\csname the#1\endcsname}%   \edef\insertframenumber{\theframenumber}%   \ifthenelse{\equal{#3}{\empty}}{%       \def\insertlistcaption{#2}%   }{%     \def\insertlistcaption{#3}%   }   \def\insertsource{#4}%   \def\insertcaption{#2}%   \ifthenelse{\equal{#1}{figure}}{%       \addtocontents{lof}{\protect\listoffigureformat{\insertcaptionnumber}{\insertlistcaption}{\insertframenumber}{\insertsource}}{}{}%   }{}   \ifthenelse{\equal{#1}{table}}{%       \addtocontents{lot}{\protect\listoftableformat{\insertcaptionnumber}{\insertlistcaption}{\insertframenumber}}{}{}%   }{}   \nobreak\vskip\abovecaptionskip\nobreak   \sbox\@tempboxa{\usebeamertemplate**{caption}}%   \ifdim \wd\@tempboxa &gt;\hsize   \usebeamertemplate**{caption}\par   \else   \global \@minipagefalse   \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%   \fi   \nobreak\vskip\belowcaptionskip\nobreak% }  \def\listoffigureformat#1#2#3#4{%   \makebox[2ex][r]{#1}%   \hspace{1ex}%   {\usebeamercolor[fg]{bibliography entry author} #2}%   \ifthenelse{\equal{#4}{\empty}}{}{ -- #4}%   \dotfill%   \makebox[2ex][r]{#3}\par% } \def\listoffigures{%   \setlength{\leftskip}{3ex}   \setlength{\parindent}{-3ex}   \@starttoc{lof}% } \def\listoftableformat#1#2#3{\makebox[2ex][r]{#1}\hspace{1ex}#2\dotfill\makebox[2ex][r]{#3}\par} \def\listoftables{%   \setlength{\leftskip}{3ex}   \setlength{\parindent}{-3ex}   \@starttoc{lot}% }  \long\def\@caption#1[#2]#3{   \par\nobreak   \begingroup     \@parboxrestore     \if@minipage       \@setminipage     \fi     \beamer@makecaption{#1}{\ignorespaces #3}{\ignorespaces #3}{\ignorespaces #2}\par\nobreak     \endgroup}  \makeatother In MySql's interpreter, it's very easy to dump a table to the screen along with its field names.   There seems to be no simple way to export a table to a tab-delimted or CSV outfile including its column headers.  I'm trying to do this using only SQL or the Linux command line, without writing a program in another language.  Thank you Faster way of writing \mathbf Beryllium Vacuum Sphere Boat/Aircraft Is it possible to make a solid rigid evacuated "balloon" out of Beryllium or other elements or alloys?  The critical buckling pressure at which an evacuated sphere is given as $$ P_1=\frac{2E\cdot\left(h/R\right)^2}{\sqrt{3\left(1-\nu^2\right)}} $$ where $E$ is , $h$ the shell's thickness, $R$ the radius to the midsurface shell, and $\nu$ is .  Of course this is theoretical and experimental and literature values for submersibles hover around 1/4 of the theoretical pressure.  Beryllium has the following properties:   Density=1.85 g/cm$^3$ Young's Modulus=287*10$^9$ Pa Poisson Ratio=0.032   Calculating the upward force, buoyancy at 101.3 kPa atmospheric pressure and Radius of 10m gives 4.24*10$^8$ N. The weight of this sphere with shell thickness 1mm has 2.32*10$^7$ N downward force of Beryllium material, so it will theoretically float at ground level.   Calculating the critical buckling stress of this beryllium sphere is 331.57 kPa, which is 3.27 times the atmospheric pressure of 101.3 kPa, so theoretically it won't buckle. But in experimentation done in earlier experiments as stated before, due to construction/material non-ideal situations, the critical buckling formula predicts about 4 times stronger than actual, so it would need about 400 kPa, not 331 kPa as Beryllium sphere with this specification would hold. Increasing the strength capability also increases weight and therefore lowers it's ability to float, but Beryllium is the closest element with this situation I have calculated that could work if ideal (I have only been using periodic table elements and household materials)  Is it possible with an alloy of elements? Using the equations for buckling and buoyancy, the following is needed. What alloys/materials come to your mind?  Material considerations:   Material with larger Poisson ratio Material with larger Young's Modulus Less dense material Construction (is it possible)   Material considerations affect on lift   Thickness can be larger, increases strength by the squared. But also increases weight. If a material with lower density with similar structural properties as Beryllium is used, it would help. Radius can be lower for same reason, but also lowers buoyancy (volume)   My other designs (don't worry about these just yet, just if you are curious is a vacuum sphere with a pressurized "outer tube" with spokes in tension (carbon fiber nanotubes maybe) to keep the 2 spheres in position. Another is a spherical truss out of carbon fiber nanotubes with a light membrane around. The truss prevents the outer membrane from collapsing due to the atmospheric pressure  The "point" of this if you ask, as to why not use a traditional balloon is it can be used as a boat by filling the sphere with air, and then used as a aircraft at will by evacuating the sphere. An alternative mode of transportation for sea vessels etc. CK2 is a big game with lots of variables, so I'll stick to adding them upon being requested to minimize the hassle.  Anyway, let's say that you have a lot of children and the oldest one claims all the land according to laws of succession. After the father dies, the circumstances will obviously piss off a lot of people (in my case that would be 6 out of 8 vassals and 4 siblings with an opinion of -50 and below one of which is a sister married to the Basileus of the Byzantine friggin' Empire, a case of excommunication, 4 failed assassination attempts under 3 years... and lots and lots of vikings). Let's also say that the current playable character is completely incompetent compare to the next sibling in line.  So, what would be the most optimal methods to off myself without further screwing over the countryside so the next heir in line has to fix it (if we assume that said heir is actually capable of doing so)?  This is my list of DLCs (not including music and unit packs):   Charlemange Legacy of Rome Sword of Islam The Republic Way of Life I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$)  to prove any possible statement? It has been ages since i studied logic at the university so i would prefer a simple explanation (if possible) Cycle Normal Pass in Camera Space Reasons to use opamp Is it possible to serialize and deserialize a class in C++? $a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$. If $a_1, a_2, ..., a_n$ is a reduced residue system $\mod p$ and $a_1', a_2', ..., a_n'$ is another reduced residue system $\mod p$ then $a_1 a_1', a_2 a_2', ..., a_n a_n'$  is not a reduced modulo system $\mod p$.  Not able to show a counter to it. Help Needed. TOC entries and labels for included PDF pages I am embedding full pages from another PDF document (using the pdfpages package), but I would like to be able to declare my own section headings before each embedded document (for e.g. hyperlinking / TOC purposes). pdfpages has an experimental feature which lets me do this, but the problem is that I want to put all of these embedded documents inside a section, so I need to insert a section heading and a subsection heading on the first document I embed. Unfortunately pdfpages doesn't seem to be able to do this. Is there a way for me to do this?  Edit: Thank you for the suggestions. I am also in need of something similar for general labels — at the moment I am abusing the addtolist option of \includePDF but the objects I'm labelling aren't really figures or tables. I have not idea how to approach this differential equation.   $$y'-\int_0^xy(t)dt=2$$.  Basically, I did,  $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$  I am stuck.  Thank You. 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. Will an object always fall at an infinite speed in a black hole? What is the name of the depiction of concentration with raised eyebrow called? Algorithm to match one input file with given numbers of file . If you filter out a range of wavelengths, how can you calculate the energy from the remaining wavelengths?   To make it concrete, say we filter everything below 450nm or alternatively above 700nm. What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists?