Closed form solution for $\sum_{n=0}^{\infty} a^{\sqrt{n}}$
Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum?
Why is there no overload of Interlocked.Add that accepts Doubles as parameters? Create a shortcut for URL? I would like to run  something &gt; file   on a remote system through ssh, but if I run  ssh host something &gt; file   the redirection is executed locally as ssh etc &gt; file  I've tried it with ' or '' or dd or with a pipe | instead, but I can't get it to work. How can this be done? Prove that $\displaystyle \sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$   $\bf{My\; Try::}$ We can write $$\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{n!}{r!\times (n-r)!}\times \frac{r!\times 3!}{(r+3)!} = \frac{3!\times n!}{(n-r)!\times (r+3)!}$$  So our sum is $$ 3!\sum^{n}_{r=0}(-1)^r\frac{n!}{(n-r)!\times (r+3)!} = \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\frac{(n+3)!}{(r+3)!\times (n-r)!}$$  So we get Sum $$ \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\binom{n+3}{r+3} = -\frac{3!}{(n+1)(n+2)}\sum^{n}_{r=0}(-1)^{r+3}\binom{n+3}{r+3}$$  $$  = -\frac{3!}{(n+1)(n+2)(n+3)}\left[-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]$$  $$=-\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]+\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}\right]$$  So we get $$ = \frac{3!}{(n+1)(n+2)(n+3)}\left[1-(n+3)+\frac{(n+3)(n+2)}{2}\right] = \frac{3!}{(n+1)(n+2)}\times \frac{n^2+3n+2}{2} = \frac{3!}{(n+1)(n+2)(n+3)}\times \frac{(n+1)(n+2)}{2} = \frac{3!}{2(n+3)}$$  Can we prove it Using Combinatorial way,bcz above method is very lengthy  If yes then plz explain here  Thanks How can I use variables in the LHS and RHS of a sed substitution? I want to do:  cat update_via_sed.sh | sed 's/old_name/new_name/' &gt; new_update_via_sed.sh   in my program.  But I want to use variables, e.g.  old_run='old_name_952' new_run='old_name_953'   I have tried using them but the substitution doesn't happen (no error). I have tried:  cat update_via_sed.sh | sed 's/old_run/new_run/' cat update_via_sed.sh | sed 's/$old_run/$new_run/' cat update_via_sed.sh | sed 's/${old_run}/${new_run}/' Find Oracle JDBC driver in Maven repository How to make a Copper/Rose Gold smudge material? I'm new to the world of Apple, I bought a new Macbook Pro, mid-2020, Intel i7, 16GB RAM. I'm a developer, I run apps like VS Code, Sublime Text, Slack, Chrome (Multiple Windows), I play videos/songs in the background. I have been keeping a track of the energy consumption on my Mac, and it only runs for 4 hours. I have reset my SMC after I upgraded to Big Sur. Is 4 hours normal? or should I go to the service center? Update on 3 Feb 2021: I submitted my Mac to the Apple service center they said they conducted a few tests and the conclusion was that the battery is fine (I don't know what type of test they did). So I guess I'll be using it on AC Update on 12 Feb 2021: Took it to the service center, they didn't find any issues so I started using Safari for work and my battery has improved by 1 hour on Average. Option + click in iTerm In Terminal, Option + click allows me to place my cursor at any point on the current line-- great for editing really long commands.  How can I do this in iTerm2?  Update: According to , the feature has been around since 0.7.0 (2003). I've also noticed that if I hold Command+Option, the cursor changes into the familiar cross like in Terminal...  but clicking still does not reposition my cursor. I am thinking of buying a digital SLR camera. What things should I be looking for? Our admins deployed a new version of Notepad++ (version 6.3) on our machines, and this version is highlighting all changes done in a file by default. It does so by setting the background color of every modified line to a bright orange, which I find very distracting.  We  have dozens of plugins installed, but I double checked and we do not have the "Change Markers" plugin (which could be the culprit).  Any idea where I can disable this feature? Changing the highlight color could also be an option. I have a question on Holder space.Is the space $C^2(\mathbb{R}^n)$ embedded in $C^{1,\alpha}(\mathbb{R}^n)$? I think in the bounded domain case, this should be true. But what if in the case where the domain is not bounded? Thanks for any hint! My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :) importing a multipage PDF file into InDesign CC When I place a multipage pdf file into InDesign CC, the pages go outside of my document settings, making this function useless.  What do I have to do to make the placed  pdf pages go within my document settings, which in this case is 11 in. Length X 8.5 in. width, .5 in. margins on all sides, with two columns per page. I would greatly appreciate an answer to this, Thank you. I am using find -type f command to recursively find all files from a certain starting directory. However, I would like to have some directories prevented from entering and extracting names of files inside. So basically I am looking for something like:  find . -type f ! -name "avoid_this_directory_please"   Is there a functioning alternative to this? What are the primary differences between the rules of D&D 3.5 and Pathfinder? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. Order of operations using Object Initializer Syntax Does the order in which I set properties using the object initializer syntax get executed in the exact same order?  For instance if I do this:  var s = new Person { FirstName = "Micah",                      LastName = "Martin",                      IsLoaded = true                    }   will each property get set in the same order? MacBook Pro stuck in gray screen when booting I've got a Late-2011 15" MacBook Pro stuck in the gray screen when booting AFTER login. I've tried removing the drive (it's non-retina) and booting from a working bootable Yosemite installer, no luck. (it sees the startup USB drive, the progress bar progresses, but stuck at the gray screen again even without a drive.  Tried NVRAM reset (cmd option P R), it has reset (the login sound's volume changed) but nothing changed.  Tried holding D on startup, AHT doesn't start, instead, it tries to boot into the OS.  Tried Cmd V, it sometimes works and displays the verbose output, sometimes, it acts like I haven't even pressed. Again, stuck after login.  Tried booting again from USB, I select USB (after seeing both drives correctly), but no luck, it's also stuck.  I've stumbled upon this:    I've tried booting into single user mode with Cmd S, it worked ONCE, I've disabled the graphics extensions just as shown at that post, restarted, but again, nothing changed.  More weirdly, now the Cmd S doesn't work either. It just takes me to login screen without showing the terminal, just as I haven't pressed anything.  Why would this happen? Why would a Mac refuse to enter hardware test mode, single user mode, or refuse to boot from a working bootable Yosemite drive both with or without the main drive installed?  UPDATE: I've run the AHT and got an error:  4MEM/66/40000000: 0x84e0ec98  It looks like a memory issue, but I'm not sure.   UPDATE 2: I've tried reinserting the RAMs and the the problem is not reported anymore. However, the Mac still won't boot. Do we still need swap partitions on desktops? I have read , but it was asked in the context of servers not desktops or notebooks.  If a good amount of RAM is installed on a notebook, say 8 GB, do we still need a swap space? How to check if a string starts with a specified string?