I would like to show that $$\lim_{n\to\infty}\sum_{k=n}^{2n}\frac{1}{k} = \ln 2.$$ Any tips on how to go about it? I feel like I should use the squeeze theorem but I fail to find convergent lower and upper bounds.
What is the sum of the 'second half' of the harmonic series?  $$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$  More precisely, what is the limit of the above sequence of partial sums?
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You can see all the ads that currently make the threshold and meet our criteria on this page:  You can also view the ads that meet the threshold using 's I am looking for idioms or informal/slang/colloquial expression for some people that make you think that they are able of building a skyscraper, constructing a spaceship, playing the piano better than Mozart or something very fancy that requires remarkable skills, but when it comes to practice they prove to be completely inadequate. When should I add another answer? I have been member of SO for 4 months. I have a question regarding some functionality. Whenever I give an answer to a question, there is functionality given Add Another Answer. I can comment on my answer at any time and also able to edit answer. Then   Why this functionality given on SO to add another answer?  When should I add another answer?  Why there is requirement of multiple answers? What is limit of adding answers to single question? How to calculate stays against 90/180 visa rules? Some destinations have a "90/180 rule" which says you can come and go but you must not spend over 90 days out of any 180 days in the area.  concerns this rule.  At least Schengen and Turkey have this rule, probably depending on where the traveller is from and what kind of visa they apply for.  Maybe I'm a dummy but I've always had a lot of trouble trying to add up how many days I've spent in such places when I've entered and left several times over recent months.  Is there an easy way to do this I'm missing or is it difficult for everybody? I'm guessing something based on counting backwards from the end date might be the trick?  Maybe you know of an app or website that can do this for you. Fastest way to remove non-numeric characters from a VARCHAR in SQL Server I'm writing an import utility that is using phone numbers as a unique key within the import.  I need to check that the phone number does not already exist in my DB. The problem is that   phone numbers in the DB could have things like dashes and parenthesis and possibly other things. I wrote a function to remove these things, the problem is that it is slow and with thousands of records in my DB and thousands of records to import at once, this process can be unacceptably slow. I've already made the phone number column an index.  I tried using the script from this post:   But that didn't speed it up any.  Is there a faster way to remove non-numeric characters? Something that can perform well when 10,000 to 100,000 records have to be compared.  Whatever is done needs to perform fast.  Update Given what people responded with, I think I'm going to have to clean the fields before I run the import utility.   To answer the question of what I'm writing the import utility in, it is a C# app. I'm comparing BIGINT to BIGINT now, with no need to alter DB data and I'm still taking a performance hit with a very small set of data (about 2000 records).   Could comparing BIGINT to BIGINT be slowing things down?  I've optimized the code side of my app as much as I can (removed regexes, removed unneccessary DB calls). Although I can't isolate SQL as the source of the problem anymore, I still feel like it is. Category-theoretic limit related to topological limit? How to create meaningful relationships with NPCs How do I help the PCs to create meaningful relationships with NPCs?   Maybe it is a result of my group playing too many one-dimensional adventures trying to find whatever MacGuffin they are after each week, but most of the time my PCs don't tend to value NPCs too much. Every time they meet an NPC they quickly evaluate what immediate values the NPC can give or tend to ignore them.  What I would really like is to have some good relationships where the PCs actually go out of their way to help NPCs without selfish motivation or even just to try to think of NPCs more and how their actions affect others.  I am not looking for a specific game rules solution to this, but instead techniques to get the players to feel attachment to these fictional characters. Support for N-Gons (polygons consisting of more than 4 vertices) has been in Blender for a while now, but many of us still may not know when to use them, and how to do so effectively.  In what situations do N-Gons really come in handy? When should they be just plain left out? Do they have any place in finished models? What's the difference between TRUNCATE and DELETE in SQL What's the difference between TRUNCATE and DELETE in SQL?  If your answer is platform specific, please indicate that. I am trying to modify output in my twig file. In D7, I was able modify data in my array with this  check_markup($tweet-&gt;text, 'full_html');   and I see that this function still exists in D8. However, I'm not sure how I would use this in a filter in twig. This is what I have so far:  {% for tweet in params.tweets %} &lt;div class="tweet-wrapper"&gt; &lt;div class="tweet-profile-image"&gt;     &lt;img src='{{ tweet.user.profile_image_url }}'/&gt; &lt;/div&gt; &lt;div class='tweet-text'&gt;     &lt;!--check_markup($tweet-&gt;text, 'full_html');--&gt;     {{ tweet.text }} &lt;/div&gt;   &lt;/div&gt; {% endfor %}   I actually tried to do this before I sent it to the template as well, but it didn't change anything:   foreach($tweets as $tweet) {   $tweet-&gt;text = check_markup($tweet-&gt;text, 'full_html');   $cleanTweets[] = $tweet; } dpm($tweets); $params = array('tweets' =&gt; $cleanTweets); $tweet_template = array('#theme' =&gt; 'twitter_pull_tweet_listing', '#params' =&gt; $params); return $tweet_template;   What I'm trying to do here is make drupal create links for web links in the twitter tweet text. This worked fine for me in D7. How to make `man` work for shell builtin commands and keywords? I am trying to install Wordpress on Lubuntu. I followed the instructions . Including PHP Apache and MySQL setups. Apache and MySQL appear to be running fine.  However, when I go to http://localhost/blog/, I get the following error:     Neither /etc/wordpress/config-localhost.php nor /etc/wordpress/config-localhost.php could be found.   Ensure one of them exists, is readable by the webserver and contains the right password/username. How prove this equation $\alpha=\aleph_{0}\alpha$ Question:     let $\alpha$ be a infinite cardinal , show that:   $$\alpha=\aleph_{0}\alpha$$   where $\aleph_{0}$ is the cardinality of the natural numbers is denoted aleph-null   this equation is from a book,and I can't prove it,and the author say this is important equation,and can't prove it.Thank you for you help macOS app for showing external HDD usage statistics? I'm planning on selling a used external HDD on eBay.  Are there any applications for Mac out there (preferably free) that can show me statistics about the usage history for the drive so that I can let potential buyers know the condition more precisely? In the Avengers movie, was Thor going to kill Captain America during his woods battle with Iron Man? How does Delta baryon decay conserve angular momentum? I'm a chemist so bear with me:  I understand the  $\Delta^{+}$ and $\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0.  I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that  into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$.  This seems to suggest momenta of:  $\Delta^{+} (J=\frac{3}{2}) \rightarrow{} p^{+} (J=\frac{1}{2}) + \pi^{0} (J=0)$  Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else. Finding descendants of British great grand uncle who emigrated to Australia? From  I have received the information of the person with the possibility of great-grandfather.   However, I have a bigger problem from here which is whether he really is my great-grandfather because there was a brother to him in the information.   If I can find his brother's descendants, then I may be able to learn the kinship with DNA testing.   How can I find the living descendants of this great grand uncle?  I was born and grew up in Japan where research is easy for me but I do not know the best way in Britain and Australia.  C.Z.Ede is the name of the candidate for my great-grandfather and his information can be found at   There is also a Mr Peter Zohrab site () which mentions "Sydney Cuthbert Ede, Eagleville Station, Talbotville, Victoria, Australia".  Mr Peter Zohrab from that site suggested that I look for descendants of Sydney Cuthbert Ede in Australia or New Zealand. However, I have not yet found any. I am also inquiring at the church in which he was baptized. How can I navigate my scene FPS style similar to how Unity does? I am aware of ShiftF fly, but it's extremely cumbersome having to use the scroll wheel to change the velocity of flight and moving the mouse from the center to spin.  Is there a way to move the viewport camera (not the scene camera) like Unity lets you right click and fly around with WASD and moving the mouse? I use Internet Explorer for some things and Chrome for others. I'd like to have a choice when I click on a link of which browser to open the link in. Is this possible in Windows? I am currently running Windows 8, although there should be similarities between versions.  To the best of my knowledge this isn't possible. How to make unit selection circles merge?