I'm trying to prove that  $$\sum_{k=0}^n {n \choose k} (-1)^k \frac{1}{k+1} = \frac{1}{n+1}$$  So far I've tried induction (which doesn't really work at all), using well known facts such as  $$\sum_{k=0}^n {n \choose k} (-1)^k = 0$$  and trying to apply identities like  $${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$$  Would anyone be able to point me towards the right method? Should I be looking to apply an identity or is there a method I'm missing?
Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method.
Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime To install the graphics driver I need to exit GUI / Gnome and go to the terminal. How can I do this in Ubuntu 18.04? Ctrl(STRG)+Alt+F1...F7 does not work, neither on the desktop, nor on the login screen... See a list of my top voted comments? Or another way to track progress on Pundit? Pundit  Left 10 comments with score of 5 or more   I saw a discussion question that mentioned Pundit, which that stated that , however, something like top voted comments seems to fit in with the rest of stackoverflows feature set, as well as indirectly provide a method to track progress on this badge. Should there be (or is there already) a list of my top voted comments somewhere? How to combine measurement error with statistic error We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times.  I know that I will have a $\Delta t = 0.1 \, \mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1.  What is the total error then? Do I add them up, like so?  $$\sqrt{1^2 + 0.1^2}$$  Or is it just the 1 and I discard the (systematic?) error of my reaction time?  I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1? Set the location in iPhone Simulator Create default home directory for existing user in terminal How do I do this? I created a user without a home directory and now I want to create one. Not just a folder called new_user, I want the default home directory with all it's folders and hidden files, etc. How do I catalog files on several external hard drives that I want to store off-line? I have quite a few external hard drives that I want to store off-line, and possibly off-site.  The contents is not saved in a structured manner, however; basically  there's lots of project data for lots of projects.  Is there any software for Windows XP that will catalog all the hard drives, so I can search the catalog, find the files, and get the ID of the disk that has the content? I have a population of phone calls - 200,000.  There are different reasons for each call, but lets assume the number of reasons is known. i.e. 7 different call reasons: 1) Check on order 2) Cancel order 3) Billing information 4) Account information etc...  My question is, I want to get a statistically significant sample size so the number of calls I listen to in my sample are representative of the distribution of the reasons in the overall population of 200,000 calls.  What should the sample size be? Which methods to use to calculate the sample size? Say I have a metric space $(X, d)$ and a set $A\in X$.  I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help? I need some help with the exercise I was assigned in math class. We were given a 3x3 square made of unit squares, and we wanted to paint all the squares. We had 3 sets of paint, red, blue, and green. Each paint was distinct from each other, but two unit squares with the same color were the same. Additionally, there's only enough paint of each color to color 3 unit squares, and all the paint has to be used. The goal of the exercise was to find the total amount of distinct possible ways such that no two unit squares that share an edge (that is to say they are next to each other vertically or horizontally, having same color across a diagonal is fine). I've started by trying to take a look at each possible arrangement of the bottom 3 unit squares, but I can't seem to find a pattern between it. Is there a better way than continuing by brute force? Any ideas on how I should proceed from here? Limits, find a limit that exist in absolute value but not outside the absolute value. Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist.   Not really sure of an example for this? Is an option missing in the new off-topic reasons? Acount suspension reason is only shown on the main site profile not meta How do you get the footer to stay at the bottom of a Web page? I have a simple 2-column layout with a footer that clears both the right and left div in my markup.  My problem is that I can't get the footer to stay at the bottom of the page in all browsers.  It works if the content pushes the footer down, but that's not always the case. How to create a Gant-Chart in LaTeX-Beamer I am trying to create a Gant-Chart in Beamer. I found the following example:   \documentclass{beamer}  \usepackage[utf8]{inputenc} \usepackage{pgfgantt}  \title{Sample title} \author{Anonymous} \institute{Overleaf} \date{2019}   \begin{document}  \frame{\titlepage} \begin{frame}{Gant-Chart}     \begin{ganttchart}[vgrid, hgrid,                         bar/.append style={fill=blue!80},                        milestone/.append style={fill=green}]{1}{15}         \gantttitle{2019}{15} \\         \gantttitlelist{1,...,15}{1} \\         \ganttbar{Task 1}{1}{2} \\         \ganttmilestone{Milestone}{7} \ganttnewline         \ganttbar{Final Task}{8}{12}     \end{ganttchart} \end{frame}    \end{document}   Now, I want to change it slightly. The first row shows the year. I want the second row to show months. And then the third row shows the number of weeks that I have planned for the task. Also, I don't want to see any arrows, but just have the rectangles. Is there a way to add a grid? Basically I want everything to just look as in an Excel-Table: Grids and then color the rectangles during which I am working.   Here is an example of what I would like to have:   Can somebody show me how to do this?   Here is another example that seems to get pretty close to what I want. Only in the week-row now it is always written "week 1", "week 2" and so on. I just want the number.   \begin{frame} \begin{figure}[h!bt]     \begin{center}         \begin{ganttchart}[             vgrid={*{6}{draw=none}, dotted},             x unit=.08cm,             y unit title=.6cm,             y unit chart=.6cm,             time slot format=isodate,             time slot format/start date=2019-02-01]{2019-02-01}{2019-07-31}             \ganttset{bar height=.6}             \gantttitlecalendar{year, month=name, week} \\             \ganttbar[bar/.append style={fill=blue}]{Task 1}{2019-02-11}{2019-02-17}\\         \end{ganttchart}     \end{center}     \caption{Time Plan} \end{figure} \end{frame} Intersection of two lists in Bash Do Ryzen 3 2200u laptops with Vega 3 work out of the box with major Linux distros? Problem starting Gnome terminal Curve Object Deformed by Curve - Distortions (Artefacts) Shaping curve object by curve (modifier) results in a terrible shape with cracked surface. Here is an extreme situation, but good visible issue.  Perspective   Top   How can I curve it smootly without adding more control points or remeshing? Please make the "last seen" entry optional Possible Duplicate:          Some people don't like being spied on by bosses or others, or revealing the frequency and intensity of their participation on SE sites.  Is an option to hide "seen" and similar fields on the summary page viable? Or, could the field be changed to day-level granularity?