How do we evaluate the sum:   \begin{equation*} 1+2+...+n \end{equation*}  I don't need the proof with the mathematical induction, but the technique to evaluate this series.
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.  How? What's the proof? Or maybe it is self apparent just looking at the above?   PS: This problem is known as "The sum of the first $n$ positive integers".
Can I travel to Paris for a day with a single entry visa for Schengen from Switzerland? Can a Half-Elf have Drow Ancestry? Checking if file is checked out through CSOM Is there a way to check if a file is checked out by another user? I am going trhough a list file by file and would like to avoid the "file is opened by another user" error. Something like this would be nice:  Guid id = new Guid(slideLibrary); // ID of Slide-Library. List list = ctx.Web.Lists.GetById(id); ListItemCollection items = list.GetItems(CamlQuery.CreateAllItemsQuery()); ctx.Load(items); // Loading all the fields System.Security.SecureString pwd = new System.Security.SecureString(); foreach (char c in "pwd".ToCharArray()) pwd.AppendChar(c);  ctx.Credentials = new SharePointOnlineCredentials(login, pwd); ctx.ExecuteQuery();  // Iterating through all the entries. foreach (var item in items) {     String currentID = item["ID"].ToString(); // Getting the ID of the current item.      if (!item.IsCheckedOut()) // This is the part I am looking for.     {        // Do stuff     }     else     {        Console.WriteLine("Item is blocked, skipping it.")     } }   I have yet not been able to find something fitting. .Net 'Any Framework' configuration delete vs delete[] operators in C++ So I recently saw add comment (requires 50 reputation) on a deleted answer. After looking around a bit, it seems to be on all deleted posts now (only deleted posts, normal posts still say just add comment).  I'm sure it wasn't there before (I think it just said add comment (unclickable) or possible nothing).  It is also on , presumably the root cause is the same.  And yes, I do have more than 50 reputation (in case anyone was wondering).  Just to clarify, requires 50 reputation obviously shouldn't be there for users with 50 or more reputation. Choices:   Display nothing Unclickable add comment - probably not ideal Unclickable add comment (disabled for deleted posts) or something along those lines   FYI - You may be somewhat confused right about now if you can't  yet. Convergence of $\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$ I am trying to conclude about the convergence of $$\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$$ If I take ${\sqrt[n]{{\sqrt[n]{n}-1 \over n}}}= {\sqrt[n]{\sqrt[n]{n}-1} \over \sqrt[n]{n}}$ this looks like it converges to $0$ but I am suspicious about the numerator. At the same time, other techniques I know like the ratio test, $n(\sqrt[n]{n}-1)$ and comparisons don't yield meaningful results. Any hints? Ease up on the Hot Network Question Randomness Whenever I refresh Stack Exchange sites or go to a question, the Hot Network Question list items are re-listed or re-ordered, quite dramatically it seems.  This seems like a .  It's actually a bit maddening to me to see the list change so frequently (or clearly I am refreshing too often or looking at it too much :-).  I understand the need to keep the list updated, but can the items in the list hold their positions a little more consistently?  The hot items can't be changing that fast. When I press Ctrl+Alt+T the Terminal opens after around 30 seconds. That's pretty annoying. If I open it with Alt+F2 and enter gnome-terminal the window appears instantly as usual.  I'm using Ubuntu 16.04 with gnome3.  Thank you for your help in advance! Why was Severus Snape placed in Slytherin? How to detect if JavaScript is disabled? There was a post this morning asking about how many people disable JavaScript. Then I began to wonder what techniques might be used to determine if the user has it disabled.   Does anyone know of some short/simple ways to detect if JavaScript is disabled? My intention is to give a warning that the site is not able to function properly without the browser having JS enabled.   Eventually I would want to redirect them to content that is able to work in the absence of JS, but I need this detection as a placeholder to start. What is the maximum number of pucture repairs an inner tube can take? What is that component from the ADP5091 Schematic? I am looking for an explanation what that component does inside of that circuit.  It can be found on page 3 of the Datasheet of ADP5091:   It would help me out a lot. Thank you in advance! Why do backgrounded processes sometimes stop spontaneously? Sometimes, some time after I've backgrounded a process with bg in bash, when I press Enter in the same shell to redisplay the prompt (just to check that I'm still in bash when some output from the background process has been displayed), the background process seems to stop spontaneously.  If I do bg again the same problem recurs.  The only way to fix it seems to be fg.  Why does this happen? Making ArcPy interactive? Confirming that my own question is a duplicate by voting to close should have immediate effect Someone voted to close my question as a duplicate. I agreed, so I cast my close vote (I have enough reputation to vote to close).  . So I expected this to happen.  But it didn't. This confused me — is this not a question I asked after all? No, it is. Maybe I need to reload the page? Nope, the question still isn't closed. Is the asker confirmation feature broken? Well, there's the “This question may already have an answer here” banner at the top. What if I click “That solved my problem!”? Ah, that does close the question.  It turns out that only clicking “That solved my problem!” instantly closes the question. Casting a close vote has a different effect, it's just an ordinary close vote.  Please make casting a close vote equivalent to clicking “That solved my problem!”. How did the house elves start serving wizards? How/Why/When did the house elves start serving wizards? I tried on my own to find the answer but I couldn't find anything relevant to my subject but I am bad at searching. WPF ComboBox without drop-down button I'm running the Storm King's Thunder campaign, and we are at the portion where the PCs get an airship to travel around in, the airship is about 1000ft in the air. There are approximately 5-6 lvl 7 PCs, and they have 3 or 4 hard encounters a day. There are two wizards, a warlock, ranger, paladin, blood hunter, sorcerer, and fighter. For the most part, they have had fun with the different weapons and enemies they have faced; however, I find that one or two spellcasting PCs have been using spells such as mind sliver, polymorph, and hypnotic pattern to defeat more challenging encounters, such as one with a roc and wyvern. They make the creature make saving throws with hypnotic pattern and cause the enemy to fall to its death, and using mind sliver makes it even harder, especially on low wisdom creatures like rocs. Or with the polymorph spell, they cause the enemy to become a fish or chicken and throw it off the edge, and since the damage carries over, it kills the monster. I realize it was legal and made fights funnier, but it does get annoying when it's happening in almost every fight. Are there any suggestions on how to keep them on track without them making every encounter easy? I thought of giving the monsters immunity to these effects, but it just sounds like a bad excuse for me not being prepared. Show that one-sided limits always exist for a monotone function (on an interval)