Find the value of $\lim_{n\to\infty}\frac{1+2^{1/2}+3^{1/3}+\cdots+n^{1/n}}{n}$ The given problem is a problem of calculus. I have tried this problem only to find that the numerator is a divergent series. No idea how to do this problem  $$\lim_{n\to\infty}\frac{1+2^{1/2}+3^{1/3}+\cdots+n^{1/n}}{n}$$
Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$. I have a problem with the calculation of the following limit. \begin{equation} \lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \end{equation} I do not know where to start! Thank you very much
Solving a system of ODEs with the Runge-Kutta method ReaderWriterLockSlim vs. Monitor I have an IDictionary&lt;TKey,TValue&gt; implementation that internally holds n other Dictionary&lt;TKey, TValue&gt; and distributes that insertions by the HashCode of the key to the invidual sub-dictionaries. With 16 sub-dictionaries, the number of collisions is pretty low on a 4-core machine.  For parallel insertions, i locked the Add-method with a ReaderWriterLockSlim, locking only the individual sub-dictionary:    public void Add(TKey key, TValue value)         {             int poolIndex = GetPoolIndex(key);             this.locks[poolIndex].EnterWriteLock();             try             {                 this.pools[poolIndex].Add(key, value);             }             finally             {                 this.locks[poolIndex].ExitWriteLock();             }         }   When inserting items with four threads, i only got about 32% cpu usage and bad performance. So i replaced the ReaderWriterLockSlim by a Monitor (i.e., the lock keyword). CPU usage was now at nearly 100% and the performance was more than doubled.   My question is: Why did the CPU usage increase? The number of collisions should not have changed. What makes ReaderWriterLock.EnterWriteLock wait so many times? Our DM keeps interrupting our long rests In the latest campaign, we're still around the third level, but every single night we've spend in the wild so far, we were interrupted by hard combat encounters (as in DMG-worded "hard"), forcing us to use our resources (like HP, spell slots, Ki, once-per-day stuff) and not restoring any of those lost the previous days. I feel like the , as it lasts four to six rounds, so we usually say that we'd like to resume our rest but the answer is always "you've been interrupted, your rest is over, you can't restart any until dusk". So the only real long rests we could have always were in town.  We've always used watch-turns, and being interrupted in the first night of the campaign, we now always spend one hour to find a safe place to rest and we make that intent very clear to the DM. But yet, kobolds, goblins, ghouls and gnolls always find us in the middle of the night, but never when we travel for ten hours during the day (which is comparable to "fast travel" in his own word). Once on the second day in the wild, we were faced to a deadly encounter (as in DMG-worded "deadly") which we survived only by fleeing. And we're kind of forced to get back to the deadly encounter, which is more than one day away from the nearest town.  We're five players and I'm the only one with PnP RPG experience... from ten years ago. I don't remember being this resourceless. We asked if it was a survival campaign and the answer was that it was only a normal introduction campaign found on the Internet that he spiced up a bit. We've also asked if we did anything wrong and the answer was that we followed the normal story.  The DM is mastering his second campaign (the first one is still ongoing, but with veteran players). The DM also plays a lot of survival campaigns as a player.  Is this situation normal? If yes, why? If not, what should I say to the DM?  Note that I'm fully aware of the "long rest" rule as written on page 186 of the PHB. This is why I encouraged my team to say that we resume our rest. (thanks Bloodcinder) What is the scientific name of this evergreen plant? This plant is found in India. It is an evergreen plant. Would love to know the scientific name. I'm asking this because my HDD (Hard Disk) has 5 years, and ultimately (this year) it gave me many headaches. Many moths ago I has Windows 7 (and 10 T.P. since april) and all was good. But after a moment, instantly, all gets reversed: veeery slow, and actions which will take a tiny time, at these moment the time seems frozen (for example, instead a 1 minute boot, it turned to 5-8 minutes boot). I formatted and installed Windows every 2 months because it was the time which runs good before it gets slooww. Seeing the abuse of privacity in W10, I changed to Ubuntu 15.04 and the things lasted more in time... But again, HDD error. Click, trying to read, click, trying to read... And the upper bar (title of window/buttons of volume, language, wifi, etc.) disappeared. The left bar (where are the app's icons) was empty and the file explorer said the resource wasn't available (or something like these error). I restarted and nothing happens. I had to press CTRL+D to boot-up Ubuntu, but after hours... the same error (can't do anything except moving the mouse/cursor).  Welcome to the routine: copy files with live-cd to a pendrive, wipe HDD and install all again. And the things was calmed again... and the errors comes again. And after 2 weeks, I had to do the same. Now my PC works fine, but yesterday I heard a few "clicks" from the HDD.  Now, I remembered this  and I came to Disks and the HDD shows a new value; normally, the problematic number was "pending sectors", always changing, but now, in "irrecoverable sectors", the number is different of zero.  I never had "reallocated sectors". Now, neither. And now, while I write this lines, I've 5 "pending s." and 9 "irrecoverable s.". If this helps, the "read error rate" is 42, 5 for "write error rate".      Very well!! While I was writing "write error rate", I heard 2   clicks. And while writing this, one click more.   And 123 "CRC error rate". So, with this information: is my HDD dying? It'll last days or months? I think this problem will kill the disk in a not very far time.  The drive is a Western Digital Green 1TB-WDC WD10EARS-00Y5B1 (80.00A80) S.N.: WD-WCAV5J301610  PS: When I boot with CTRL+D, many times the system is in "RO mode"; I can't save/download files, Unity doesn't find apps when I write their name {click again, now, while I write the text} and, after a few hours, a crash like the lines above How do I configure Nautilus to open at my Documents folder instead of my home folder?  I've scoured this site and found answers instructing users to edit the /usr/share/applications/nautilus.desktop file, replacing %U with the path to the desired default directory. I've edited all the variants of nautilus.desktop in the said directory, but none of the edits has an effect. I've checked all the files after editing them to make sure that I saved the edit effectively, and all look good to me.   I've edited every file to read as follows:  Exec=nautilus --new-window /home/judy/Documents   I've also edited the org.gnome.Nautilus.desktop file, just to be sure, with no joy.   I'm running Ubuntu 15.10.  Note: if I enter nautilus /home/judy/Documents from the command line, Nautilus properly opens in the desired directory.  Is there a simple way that I can configure an icon in the launcher bar to run this command and have the Nautilus icon that looks like a metal file drawer?  Thanks. I'm pulling my hair out over this one. glob() - sort array of files by last modified datetime stamp I am trying to draw a cantor set in LaTeX and I am having issues doing this. I am trying multiple \put commands, but they just go to the right of one another. Does anyone know how to do this?  Any help would be much appreciated. Assign default keyboard language per-application Is it possible to assign default keyboard language for specific applications? I know that there is an option to always start an application with a default keyboard language but this is not what I want.  I'm looking for something similar to "fixed workspaces" which you can set using CompizConfig (You set Chrome to always open at X1Y1, terminal at X2Y1 etc). Something where I would set Chrome: Czech, terminal: English, Spotify: English, ... How can I tell hotels exist that aren't on the booking web sites? There are a ton of sites out there to help book hotel rooms and to aggregate all the different sites to book hotel rooms. These companies make enough money to buy ads on TV, so I won't name them here. I normally poke around 3 or 4 of these as part of planning any trip.  I am now starting some very long term planning for a 6 week South Pacific trip, which includes undertstanding what hotels costs on some very small islands. I have paper guidebooks and I've been looking online. But I come across anomalies. For example, a place which has only 9 hotels listed on the sites I use, and prices range from $1000/night down to $200/night, then when I read the reviews on that cheapest one people are saying "it's nothing special, there are 5 or 6 places just like it here for half the price". Well, not on this site there aren't!  My guess is that many of these places don't have web sites, or don't have online booking, and therefore aren't getting into the aggregators. For some islands I have found local tourism boards with lists of "pensions" and such, typically cheaper than what I can find on the aggregators.    But how can I know? I don't want to assume that popular web sites represent reality, and choose only from places listed there. But I don't want to invest 10-20 hours of research per island only to discover the aggregators had everything. How can I know where the coverage is good and where it isn't, what kinds of hotels are more or less likely to be listed at expedia, trivago, booking, hotels, tripadvisor, and the like? (For example, I don't want to stay at a hostel, so I don't mind if they're not included in online sites.) Are there sites like wikitravel that list places without much of an internet presence?  I'm starting my research over a year before the trip, but wading through hotel pages is not the fun part of the planning so tell me how the smart people do it. Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$. Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2}  \right |\leq \left |b-c  \right |$. Give me a clue, I don't have any idea to work with that inequality. Am I using beta or stable? I have been doing so many sudo apt-get update/(dist-)upgrade since yesterday and  something happend - so that I am confused now whether I am using still beta or the stable version.   How do I figure that out? Is Highschool of the Dead on hiatus again? Equivalent of cURL for Ruby? Are all eigenvectors, of any matrix, always orthogonal? How to write a unit-test / test class for trigger? How do I write a unit-test / test class for a trigger?  What are my options?   Inline with the trigger?  New Class? I would like to understand how Physicists think of space-time in the context of String Theory. I understand that there are $3$ large space dimensions, a time dimension, and $6$ or $7$ (or $22$) extra dimensions, and all these dimensions need to fit together in a way such that the extra dimensions are compactified (with a Calabi-Yau or $G_2$ structure).   My question, however is not about the possible $10$, $11$ or $26$ dimensional manifolds that may be possible, but about whether string theorists consider space-time as somehow quantized (or discrete), or rather as a continuous manifold, or are both options possible? In other words, can strings move continuously through space, or are there a discrete set of locations where strings can be, and does string theory rule out one of the options?  How about the same question in loop quantum gravity (LQG)? Should I think of the spin networks in LQG as describing a discrete space-time?  Thanks for your insight, or any references you may be able to provide! Beamer - Color of header foreground with AnnArbor theme I use beamer with the theme AnnArbor and colortheme whale.  I tried to use:  \setbeamercolor{section in head/foot}{fg=white, bg=blue}   or  \setbeamercolor{palette tertiary}{fg=white, bg=blue}   But for an unknown reason, the bg works but not the fg... The sections names stay black in the header.    EDIT : the problem comes from the hyperref package.  This code works well :  \documentclass{beamer} \usetheme{AnnArbor} \usecolortheme{whale} \setbeamercolor{frametitle}{fg=white,bg=blue}  \begin{document}     \section{Test 1}     \subsection{Test 1.1}     \begin{frame}         \frametitle{Test}     \end{frame} \end{document}   Whereas this one doesn't works :  \documentclass{beamer} \usetheme{AnnArbor} \usecolortheme{whale} \setbeamercolor{frametitle}{fg=white,bg=blue} \usepackage{hyperref} \hypersetup{     colorlinks=true,     linkcolor=black,     urlcolor=blue,     citecolor=gray,     frenchlinks=false,     pdfborder={0 0 0} }  \begin{document}     \section{Test 1}     \subsection{Test 1.1}     \begin{frame}         \frametitle{Test}     \end{frame} \end{document}   And that's pretty normal because of this line :  linkcolor=black, Import PLY doesn't display vertex colors I'm trying to import a .PLY mesh in Blender. I've troubles with colors: the mesh appear in a grayscale palette. Other softwares open it correctly (eg. MeshLab).  This is the header of the PLY file:  ply format binary_big_endian 1.0 comment created by Geomagic Sculpt Exporter. element vertex 182488 property float x property float y property float z property uchar red property uchar green property uchar blue element face 364980 property list uchar int vertex_indices end_header   Any tips? Thanks! En Dashes & Hyphens in Same Phrase Without recasting and in lieu of unsightly suspensive hyphenation, could these work? The en dash (which means "to") is used between the figures, and the hyphens are used in the compound modifier. To me, the examples below are concise and clear.  Do you agree that the examples below are 100% punctuated correctly--yes or no?  45–50-year-old men  10–20-mile radius  $45–$55-million-per-year industry  $100,000–$150,000-a-year savings  10–15-degree temperature difference  $2–$5-per-day service fees  45–75-cent-a-week raises  55–65-mph speed zone  15–20-ounce steaks  20%–40%-a-year tax increases