Real analysis - Series
Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges
using trial and error in math problems Suppose one bacterium is in a jar at 12AM, and then suppose there are two bacteria at 12:30AM,  and then there are four bacteria at 1AM, etc.  (a)  How many bacteria will be in the jar at 12PM that day?  (b)  Use trial and error to estimate at what time there will be 1 billion (1,000,000,000) bacteria in the jar.  for a) is it going to be 44 bacterias ? or am i wrong? Please help with b), i have no idea how to figure it out in short term Thanks Very new player here. Looking at the stat block of a goblin: for a shortbow attack it says +4 to hit and the goblin has a Dex modifier of +2.   Does this mean I add 6 to the d20 roll or just the 4 and ignore the Dex modifier?  Same with using Stealth. It says Skills: Stealth +6. So would I take 6 and add +2 of the Dex modifier or just add 6 to the d20 of the Stealth check roll?  Thank you! How to force a node group input field to display a slider? I have created a couple of node groups with some "group-inputs-sliders" (The one called "Roughness" in the bottom picture). I would like it to have the blue-bar slider like the blender native ones. Thanks (: How to set the font for a \section title (and chapter etc) How do I specify the fonts, and general font size, for the titles, chapters and sections in my document  Update:  I can see how my question is confusing.  I use \section and \chapter etc. to organise my document, and I would like to change the way the section and chapter titles are displayed. How to suppress Terminated message after killing in bash? How can you suppress the Terminated message that comes up after you kill a process in a bash script?  I tried set +bm, but that doesn't work.  I know another solution involves calling exec 2&gt; /dev/null, but is that reliable? How do I reset it back so that I can continue to see stderr? drawing at least 90% of colors from urn with large populations List files sorted by the number of lines they contain How can I list the number of lines in the files in /group/book/four/word, sorted by the number of lines they contain?   ls -l command lists them down but does not sort them Maya is sister of dhara who is doctor. Maya is sister of dhara, who is doctor.   In both statements who is used for which person??? I would like to filter the markers on my (geojson) layer according to two features: year and type of event, ideally using both filters (year and eventType) as check-boxes without using layerGroup at all. I would like to be able to click on a checkbox and update dynamically the markers by filtering them according to feature.properties.eventType too, in addition to the year displayed, something like    &lt;input type="checkbox" id="all" name="all" value="all"&gt;   &lt;label for="all"&gt;All&lt;/label&gt;   &lt;input type="checkbox" id="funfair" name="funfair" value="newsletter"&gt;   &lt;label for="funfair"&gt;Funfair&lt;/label&gt;   &lt;input type="checkbox" id="trade" name="trade" value="trade"&gt;   &lt;label for="trade"&gt;Trade&lt;/label&gt;   &lt;input type="checkbox" id="exhibition" name="exhibition" value="exhibition"&gt;   &lt;label for="exhibition"&gt;Paid exhibition&lt;/label&gt;   etc... which would trigger a  filter: function(feature, layer) {   return feature.properties.eventType == value-of-checkbox; }   I have five types of eventType. So, for instance, if 1890 is checked, the map is showing all 1890 events, no matter what kind of event it is (because by default it would show all events). If the user checks the 'funfair' and the 'trade' checkbox, though, I want the map to show only the 1890 events of the funfair and trade kind, and not other kind of events.  This is my complete script, which so far is working fine with filtering by year.  var oldmap = L.tileLayer('address', {   maxZoom: 12 }),  var modernmap = L.tileLayer('https://api.tiles.mapbox.com/v4/{id}/{z}/{x}/{y}.png?access_token={accessToken}', {   maxZoom: 12,   id: 'mapbox.streets',   accessToken: 'mytoken' });  var map = L.map('map', {   center: [40.1857864,-4.5536861],   zoom: 7,   layers: [oldmap, modernmap] });  var baseMaps = {   "Old map": oldmap,   "Modern map": modernmap };  var jsontest = {"type": "FeatureCollection",   "features": [     {       "type": "Feature",       "geometry": {         "type": "Point",         "coordinates":  [ -16.5471268,28.4136726 ]       },       "properties": {         "Name":"Point1",         "year":'1892',         "eventType":"funfair",         "notes":""       }     },     {       "type": "Feature",       "geometry": {         "type": "Point",         "coordinates":  [ 2.1753895,41.3767925 ]       },       "properties": {         "Name":"Point2",         "year":'1890',         "eventType":"trade demonstration",         "notes":""       }     }   ]};  var mapLayerGroupsYear = [];  function onEachFeature(feature, layer) {   var year = mapLayerGroupsYear[feature.properties.year];   if (year === undefined) {     year = new L.layerGroup();      //add the layer to the map     year.addTo(map);      //store layer     mapLayerGroupsYear[feature.properties.year] = year;   }    //add the feature to the layer   year.addLayer(layer);    }   var myLayer = L.geoJSON(jsontest, {   onEachFeature: onEachFeature })  L.control.layers(baseMaps, mapLayerGroupsYear).addTo(map);   I am a beginner with leaflet so please forgive the dumb question, in case it is. Why does Type I error always occur in a NHST? Or, is NHST too weak to tell the truth? Recently I have noticed that Type I error of a test is too much persistent. Even if all the values are almost equal (variance is extremely low) and tested to that value (true mean), still there is nearly 5% Type I error of the test. I am frightened and curious to rethink, are the people who have banned (or skeptic about) NHST (null hypothesis significance testing) truly correct? Let's have a look at the following r code:       n=10000 t1err=0 set.seed(1235) for (i in 1:n){    x=rnorm(100, 10, 20)    if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)  } cat("Type I error rate in percentage is", (t1err/n)*100,"%")   It will produce roughly 5% as Type I error. Seems ok. But let's see this:      n=10000   t1err=0   set.seed(1235)   for (i in 1:n){      x=rnorm(100, 10, 0.000001)      if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)    }   cat("Type I error rate in percentage is", (t1err/n)*100,"%")   Each values are extremely close to 10. Still it produces about 5% Type I error. What conclusion we may draw from it? Or, am I missing something? SHOWPLAN permission denied when including execution plan in Data Explorer If I select the "Include execution plan" checkbox in the Data Explorer, the query fails:     SHOWPLAN permission denied in database 'StackOverflow'.     I see that  following maintenance. I don't know whether there was any recent maintenance. It's been several days since the data was updated and longer since an update was deployed, if that's related. Is it ethical to use a recommendation letter from a deceased professor? She was a really nice professor and she wrote me a good recommendation letter that resulted in a lot of good results with applications. Recently, this professor died from brain cancer, which was a fast and shocking death for me.  Tomorrow is the deadline for a grant. What should I do? Is it professional to use letters from the deceased? Macroscopic quantum phenomena and tunneling Do people sometimes get blocked by a webservice for things someone else has done with their dynamic IP-Address It is common practice for operators of web services (e.g. web sites) to block IP addresses when there was malicious traffic originating from that address.  It is also common practice that internet users don't have a static IP address but get one assigned by their ISP via DHCP.  Now I would expect that it sometimes happens that a specific IP address gets banned from a service and later that address gets assigned to another user, leaving the new user (temporarily) banned from the service for no apparent reason. However, I never had this problem or heard of someone having that problem.  Does this sometimes happen? If not, what prevents this from happening? This is a  about DNS server administration.   I have one hundred or so domains. All of these domains need to be configured identically, but it seems like a huge waste of time to have to configure a new zone and/or zonefile for every one of these domains. There has to be a better way to automate this!  I think I'm on to something...if I create a zone called ., or use some other feature in my DNS software to always return a specific IP when an A record is requested, this seems to get me pretty close to my desired end result. My server is responding authoritatively to the requests and it's so much easier to manage!  This was working great until nameserver validation software started checking these domains. I figured out that I can make most of the errors go away by adding NS records, but my software won't let me put more than one SOA record in the same zone file.  How do I work around this multiple SOA record problem? I use Stack Overflow a lot and have noticed a certain trend that I myself got caught up in at one time of using the phrase "I am having a problem" in place of "I have a problem."  I would use this phrase for an event in the future, for example, "I am having a steak dinner tonight," but for a problem that I need solved and am asking for a solution, I'd use "I have a problem..."   What would force us to use this phrase? Is it even valid English for the present tense? Libraries module. Where to extract? Where should I be extracting the  module, sites/all/modules or sites/all/libraries? Which registry keys are impacted with Xerox Printer driver installation? I am implementing a driver deployment with Samba.  Every possible model of printer found an easy solution, except for Xerox Workcenter.    After browsing the Internet, I found that Xerox printer driver has to change some registry keys in order to work, keys which are changed by the executable downloadable on the manufacturer website.  Do you know which key are impacted by the Xerox Workcenter GPD driver?  This corresponds to the error message I get while uploading to Samba, which asks the position of a file named as a registry key.  I browse Samba bug tracker and they did not specify which registry key are impacted.  I contacted Xerox itself, but despite that fact that we have a support contract and really a huge number of workstations, it seems that Xerox is not willing to help us. What fallacy dismisses problems by presenting "bigger" problems? Wasn't really sure how to phrase this, but I'm thinking of an instance in which someone diminishes a problem by presenting one of larger scope - as a rather shoddy example, "x political problem in America doesn't matter because half the world's population is starving."  I honestly have no idea if this is considered fallacious or not, but for some odd reason I can't shake the feeling that someone has described it to me as a fallacy before. Am I crazy or is there something to this (or something similar)? Say I have two strings,  String s1 = "AbBaCca"; String s2 = "bac";   I want to perform a check returning that s2 is contained within s1. I can do this with:  return s1.contains(s2);   I am pretty sure that contains() is case sensitive, however I can't determine this for sure from reading the documentation. If it is then I suppose my best method would be something like:  return s1.toLowerCase().contains(s2.toLowerCase());   All this aside, is there another (possibly better) way to accomplish this without caring about case-sensitivity?