Ι got a feeling that $$\sum_{x=1}^{\infty}\Big\lvert\sum_{k=0}^{\infty} \frac{x^{2k}}{(k+1)!}(-1)^{k} \Big\rvert \geq \sum_{n=1}^{\infty} \frac{1}{n} $$  because $$\sum_{x=1}^{\infty} \Big\lvert x-\frac{x^2}{3!}+\frac{x^4}{5!}... \Big\rvert \geq 1+\frac{1}{2}+\frac{1}{3}...$$ i feel that somehow terms will get canceled but i cant prove it!!   **this came up as a part of problem i was solving . I got no idea if the above inequallity is true **
It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
Why does `find` not delete the files it found? I'm trying to understand why the find command is not deleting files in one particular case. I've got the following command:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7   Which returns the files I would expect:  C:\path\to/Pictures/pmcctv/cap_20160915T193251_620067800.jpg C:\path\to/Pictures/pmcctv/cap_20160915T193322_742708800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081046_394767500.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081125_615129600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081129_503678200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081255_842394600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161008_693586800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161012_749396800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161043_774132200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161150_497251900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161154_437379600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161337_350955300.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161341_452596000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161412_870774000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161443_969064400.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161554_468109900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161558_378086000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161916_668433100.jpg   Now if I try to run the same command with -delete, the files are not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -delete   I've also tried with -exec rm, but they are also not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -exec rm {} \;   find returns no error and the files are owned by me with permissions 644.  Any idea what could be causing this?  (note: I've cut the commands to make it more readable but normally there's no line break) Earlier today, I was suspended from chat due to an inappropriate message (I totally deserved it, but that's not the point). I looked at my phone, and this is all that it told me:    ...which confused me because chat was still moving as normal. I didn't realize that I had been suspended until I checked on my computer to see what the heck was going on.  To me, "This room is read-only" sounds like some sort of database lockup or something related specifically to the room, not the user. It doesn't give any feedback to indicate that the room will be "read-only" for 30 minutes, and doesn't mention that all other rooms are also "read-only". Even a message saying just "You have been suspended from chat" would be more helpful. I want to prove that: $\ln(x+1)&lt; x$.  My idea is to define:  $f(x) = \ln(x+1) - x$, so:  $f'(x) = \dfrac1{1+x} - 1  = \dfrac{-x}{1+x} &lt; 0, \text{ for }x &gt;0$.  Which leads to $f(x)&lt;f(0)$, so $\ln (x+1)-x&lt;0$.  Is that a valid proof? Any other ideas?  Thanks. How do I unset an element in an array in javascript? Is their a way to move multiple vertices gradually at a time? I have moved multiple vertices gradually but a set of vertices at a time not all at a time, as shown in the animation below (First I moved the four vertices from top, then three, then two and at last the last vertex).  I wanted to do the same thing as shown in the animation below but all at a time. Why do some Unicode text messages crash the iPhone OS? I read a collection of Sci-Fi short stories back in the 1970's or early 80's that had one about a couple in love who kept losing each other. The book was my dad's so the story itself could be much older than that. I am very fuzzy on the details as I was young when I read it and it was so long ago. I believe the woman/girlfriend was "taken" (by some bad guy aliens??) while she was sleeping. I think it involved time travel and the man had to search through time to find her. After he found her, when he was asleep one time, he somehow was then "taken" and I think transported through time and had to fight his way back to her again. He may have been some sort of police officer. I just remember in the end, at night when they went to bed, the couple always wore handcuffs so they were never separated when they went to sleep ever again. Does this ring a bell with anyone?  (There is a possibility it is in a short story collection that also contained " The Twonky " by Lewis Padgett (1942), as I remember reading that story around the same time) How do I logout from a non-registered account? I have a registered StackOverflow account which I normally log into using OpenID.  Recently I logged out in order to post a question anonymously (supplying a fake name and of my secondary email addresses).  The problem now is that I'm logged in to StackOverflow as this "dummy" user, unable to see a way to log out.  The top menu normally has a "logout" link, but when you aren't fully registered this link is replaced with "register".  There is no "logout" link for me. :(  So..   When logged out there is a "login" link. When logged in there is a "logout" link. When "logged in" with an unregistered account, there is only a "register" link (no "logout"). At this time, Wubi does not work with Windows 8 default boot-loader. Thus at this point Wubi would not work on a new Windows 8 machine.      Citation fom    Question  Recently I've learned of , a way to install Ubuntu right from Windows.  Besides installing from Windows, I would like to know key differences between a regular "alongside Windows" or Dual-boot Ubuntu installation and one done with Wubi. Are there any disadvantages (for example a performance penalty) or incompatibilities I should look out for when using Wubi? Creating a textarea with auto-resize There was , which I've tried. But there is one problem: the textarea doesn't shrink if you delete the content. I can't find any way to shrink it to the correct size - the clientHeight value comes back as the full size of the textarea, not its contents.  The code from that page is below:  function FitToContent(id, maxHeight) {    var text = id &amp;&amp; id.style ? id : document.getElementById(id);    if ( !text )       return;     var adjustedHeight = text.clientHeight;    if ( !maxHeight || maxHeight &gt; adjustedHeight )    {       adjustedHeight = Math.max(text.scrollHeight, adjustedHeight);       if ( maxHeight )          adjustedHeight = Math.min(maxHeight, adjustedHeight);       if ( adjustedHeight &gt; text.clientHeight )          text.style.height = adjustedHeight + "px";    } }  window.onload = function() {     document.getElementById("ta").onkeyup = function() {       FitToContent( this, 500 )     }; } I have come across both the phrases - 'on the internet' and 'over the internet.'   I'm pretty sure that both are correct but then which to use when?  Please note, I'm well aware of all over the Internet which I'm not concerned about. Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$   My (very limited) attempt so far: after creating a small list of the values $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21, F_9=34, F_{10}=55$ i can see that yes it does seem to work for instance $F_{6+3}=F_5 F_3 +F_6 F_4 = 10 +24 = 34 = F_9$. However, I really don't know where to begin as showing that this must hold in general terms. Should I be looking to use limits? Or perhaps induction? What is the best way to solve this? Female main character in an ice world earning gems/runes How do I mirror the faces of a cube (edit all of them when I edit one)? I don't think I've seen any answers to this question, but I'm curious as to why, when you have a 64-bit OS installed (ex: Vista x64), IE defaults to the 32-bit version? Are there drawbacks to using the x64 version of IE (IE8, specifically)? How do I change my Stack Exchange password? How do I change my password?  I have a Stack Exchange account but can't seem to find the option to change my password.  Help please? How can I report bugs with or request features for the Stack Exchange mobile apps? I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions. Does there exist a smooth approximation of $x \bmod y$? I'm looking for a function $m(x,y)$ that smoothly approximates $x \bmod y$, and I'm assuming there would be some $n$ or $\varepsilon$ in the body of $m(x,y)$ that defines the degree of approximation such that as $n$ goes to infinity or $\varepsilon$ goes to zero,  $$ m(x,y)= x \bmod y. $$  I've looked at, for example, $\sin(x/y)$ and have been trying to figure out ways to make $\sin(x)$ more "steep" so that I can apply such techniques to $\sin(x/y)$ in order to perhaps more closely approximate $x \bmod y$.  The analogy I'm looking at is the approximation of $\mid x\mid$ as $\sqrt{x^2 + \varepsilon)}$ where epsilon is some small quantity.  Does a similarly smooth function exist for $x \bmod y$? Does being denied entry to a country affect future travel to other countries? I was denied entry to India due to a visa technicality. I was advised by organisation I was travelling with to get tourist visa when I actually needed miscellaneous visa. Will this affect future travel to other countries if I have been refused once by a country, or if I want to re-enter India? In the story I'm trying to recall, everybody on their thirteenth birthday (?) is assigned by some kind of brain analysis method a profession and given some kind of pill (?) that instantly (?) grants them the required knowledge and skills to start working immediately. Some professions are more sought after than others, especially in view of the olympic (?) games regularly held on a profession-by-profession basis.  The protagonist, however, is turned down from this system and dismissed with some general reason and has to work his way to adulthood the boring old way: studying from books. Eventually he rejects from his status as a "special" student and goes on the run, hiding from the teachers and envying the normal people.      Eventually, it turns out that the protagonist was actually smart enough that being spoonfed premade information was a waste. What he actually was assigned to was advancing the bundles of knowledge themselves, as well as being front line in the technological progress.  "Somebody has to write those textbooks! You can infuse knowledge about how to use today's technology, but you can't do that for tomorrow's - people that get assigned a profession become tied with their assigned technology and are only as relevant as that. If you want more, you must learn how things actually work, and there's no shortcut for that." (not an actual quote, but a specific passage as I remember it.)  The troubles the protagonists underwent were actually planned and necessary for him to acknowledge and accept his role in society.   This story was certainly longer than a couple pages, but it wasn't very long either. If it's a short story, it was probably written by Asimov, but I can't be sure. It rings me of Brave New World given the theme of government-issued population planning, but it certainly doesn't fit with anything else in the actual plot. Any idea?