Is there a closed form expression for the following sum:   $$\sum_{k=0}^s (-1)^k {n \choose k}, $$  where $s \in \{0,1,2,...,n\}$  -- which is basically the first $s$ terms in the alternating binomial coefficients series.   I know if $s=n$, the sum is zero, but I cannot figure out a general closed form expression in terms of $s$.
It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem.  I'm trying to figure out what happens with the truncated sum $\displaystyle\sum_{i\ =\ 0}^{D}\left(\, -1\,\right)^{i}\binom{n}{i}$.  How far away from $0$ can this get, as a function of $D$ ?.  I'm mostly interested in the case of when $D \ll n$, such as $D \sim \,\sqrt{\,n\,}\,$.  Thanks !
I read the following dev email  and when i read this part:  Note that this does not completely remove the ability for extensions (e.g. Dropbox) to add emblems programmatically by using the libnautilus-extension library, just the user-added emblems from the properties dialog/emblems sidebar.   There is ability to bring emblems programmatically. So, How can i use libnautilus-extension library to add some emblems like Dropbox does? Or is there any tweak tool for this? File transfer using sockets (C#) - received file doesn't contain full data Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$ Given $m \in \mathbb{Z}$, let $m\mathbb{Z}$ denote the set of integer multiples of $m$, i.e. $m\mathbb{Z} := \{mk\mid k \in \mathbb{Z}\}$.   Now let $a,b \in \mathbb{Z}$ with $a,b$ not both $0$. Prove that $a\mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb{Z}$.   I am trying to write a proof for this, but I am unsure of what method to use. Also I am confused by $mk\mid k$, because wouldn't $m=1$ for this to be true. Can I turn off macOS Catalina's internal display without close the lid while using an external monitor? I'm trying to set an RSA key as an environment variable which, as a text file, contains newline characters.  Whenever I attempt to read from the file and pass it into an environment variable, it will just stop at the newline character on the first line. How can I prevent this? Margin top in inline element Difference between processing.runalg() and general.runalg() In the discussion from a question I asked several days ago (), someone suggested that I use general.runalg() instead of processing.runalg() for running QGIS algorithms.    I've now used both, and haven't noticed a difference in either performance or syntax between them.  I'm just wondering what sort of differences there are between these two methods, and which one I should be using. have lived vs have been living How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? Java Generics: List, List, List How to add thickness to a curve? How can I add thickness to a curve? I already tried with the extrude and depth options in the curve settings, but it doesn't work properly. xdg-open on debian 9 fails to open browser I decided to try lxdm (was using fluxbox and xfce), and discovered that for many programs the url handler was failing, producing this error message;   Quite strange as you can see, it's prepending the user directory to the url.  The example here is from telegram, but it happens in discord, as well as when executing from the command line; xdg-open https://www.google.com produces a similar error. xdg-settings get default-web-browser output's firefox.desktop which works as a link in both xfce and lxdm.  More information; I ran bash -x on it and...  $ bash -x /usr/bin/xdg-open http://www.google.com + check_common_commands http://www.google.com + '[' 1 -gt 0 ']' + parm=http://www.google.com + shift + case "$parm" in + '[' 0 -gt 0 ']' + '[' -z '' ']' + unset XDG_UTILS_DEBUG_LEVEL + '[' 0 -lt 1 ']' + xdg_redirect_output=' &gt; /dev/null 2&gt; /dev/null' + '[' xhttp://www.google.com '!=' x ']' + url= + '[' 1 -gt 0 ']' + parm=http://www.google.com + shift + case "$parm" in + '[' -n '' ']' + url=http://www.google.com + '[' 0 -gt 0 ']' + '[' -z http://www.google.com ']' + detectDE + unset GREP_OPTIONS + '[' -n LXDE ']' + case "${XDG_CURRENT_DESKTOP}" in + DE=lxde + '[' xlxde = x ']' + '[' xlxde = x ']' + '[' xlxde = x ']' + '[' xlxde = xgnome ']' + '[' -f /run/user/1000/flatpak-info ']' + '[' xlxde = x ']' + DEBUG 2 'Selected DE lxde' + '[' -z '' ']' + return 0 + case "${BROWSER}" in + case "$DE" in + open_lxde http://www.google.com + pcmanfm --help -a is_file_url_or_path http://www.google.com ++ file_url_to_path http://www.google.com ++ local file=http://www.google.com ++ echo http://www.google.com ++ grep -q '^file:///' ++ echo http://www.google.com + local file=http://www.google.com + echo http://www.google.com + grep -q '^/' ++ pwd + file=/home/nesmerrill/.local/share/applications/http://www.google.com + pcmanfm /home/nesmerrill/.local/share/applications/http://www.google.com + '[' 0 -eq 0 ']' + exit_success + '[' 0 -gt 0 ']' + exit 0   The important part seems to be pcmanfm --help -a is_file_url_or_path http://www.google.com but, that command if that's how it was used, doesn't seem to do much of anything?  $ pcmanfm --help -a is_file_url_or_path http://www.google.com Usage:   pcmanfm [OPTION…] [FILE1, FILE2,...]    Help Options:   -h, --help                   Show help options   --help-all                   Show all help options   --help-gtk                   Show GTK+ Options  Application Options:   -p, --profile=PROFILE        Name of configuration profile   -d, --daemon-mode            Run PCManFM as a daemon   --no-desktop                 No function. Just to be compatible with nautilus   --desktop                    Launch desktop manager   --desktop-off                Turn off desktop manager if it's running   --desktop-pref               Open desktop preference dialog   --one-screen                 Use --desktop option only for one screen   -w, --set-wallpaper=FILE     Set desktop wallpaper from image FILE   --wallpaper-mode=MODE        Set mode of desktop wallpaper. MODE=(color|stretch|fit|crop|center|tile|screen)   --show-pref=N                Open Preferences dialog on the page N   -n, --new-win                Open new window   -f, --find-files             Open a Find Files window   --role=ROLE                  Window role for usage by window manager   --display=DISPLAY            X display to use My case here is that I was writing something for school, and my teacher explained an error which I really can't see. She referred to articles about &quot;dangling participles&quot;, which was previously unknown to me. I find it hard to relate the examples I've found about it to my situation. So this is a text I've written and the bold part is where the problem arises: (You could read only the bold part if you please)  Wandering, and lost in his thoughts, there is a deep despair occupying the mind of this man. He is barely aware of his surroundings, except for those few moments when the loud blissful voices break through the protective barriers, developed by his immense sadness and grave frustration, into his mind. ”How can they all possibly be so happy?” he thinks to himself. Staring blindly, with his head low, as he walks in a slow pace, whispering ”Mary, Mary, my Mary,” over and over again. Yesterday was her funeral. She had passed away from a severe type of cancer, at a relatively young age. It was his wife. They had been together since high school and were greatly attached to each other. Having spent nearly half his life by her side, we cannot but imagine his grief. Indeed, he could hardly picture a life without her. So, alone he roams in the midst of all the seemingly incessant cheering and noises, drowned by his sorrow.  Her comments on that part was:  we is slightly out of place here – see my comment below These few sentences are about him and his emotions, so ‘we’ seems out of place. If the sentence starts: ‘Having spent nearly half his life by her side’, the reader expects it to continue: ‘he’. After all we have not spent half our lives by her side.  Is she right? If so, could anyone try to explain why? Proof of when is $A=X^TX$ invertible? apt-get install vs pip install I am a bit confused about the cases in which the above commands must be used when downloading python packages. I was trying to download a package named pyudev in accordance with an answer with this . I executed this command :  sudo pip install python-pyudev   but received the following message :  Downloading/unpacking python-pyudev    Could not find any downloads that satisfy the requirement python-pyudev Cleaning up... No distributions at all found for python-pyudev Storing complete log in /home/vineet/.pip/pip.log   However the following worked fine :  sudo apt-get install python-pyudev   When is apt-get supposed to be used to install packages and when is python-pip used? How to disable Excel number autoformat How slow are .NET exceptions? I don't want a discussion about when to and not to throw exceptions. I wish to resolve a simple issue. 99% of the time the argument for not throwing exceptions revolves around them being slow while the other side claims (with benchmark test) that the speed is not the issue. I've read numerous blogs, articles, and posts pertaining one side or the other. So which is it?  Some links from the answers: , , . Javascript - Change html tag type How do I edit an invalid sudoers file? It throws the below error and it's not allowing me to edit again to fix it.  Here is what happens:  $ sudo visudo &gt;&gt;&gt; /etc/sudoers: syntax error near line 28 &lt;&lt;&lt; sudo: parse error in /etc/sudoers near line 28 sudo: no valid sudoers sources found, quitting DataGridTextColumn.IsReadOnly seems to be faulty