Does the series ${a_n}$ = $\sum _1^n \frac{1}{n^{1+\alpha}}$ converge for all $\alpha$ > 0?
To prove the convergence of the p-series   $$\sum_{n=1}^{\infty} \frac1{n^p}$$  for $p &gt; 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.  I am wondering if there is a self-contained proof that this series converges which does not rely on either test.    I suspect that any proof would have to use the ideas behind one of these two tests.
A case of optional “that”: “check the” vs. “check that the” Consider the following use case:   Please check the username and password are correct. Please check that the username and password are correct.   In this case, I would say that that is required because it feels more natural to me and so the flow is better. However, is the lack of that wrong in this case?  The question boils down to: when is that optional? See . A quick summary:   That is optional if the pronoun is the object. That is required if the pronoun is the subject.   I think that we fall into the second case: that is required because the relative pronoun is the subject. But I'm not sure if I'm interpreting these rules correctly. Can someone please advise? Can I travel to UK with my Italian national daughter and husband with my permesso di soggiorno? I have an Italian permesso di soggiorno and I want to join my husband in the UK. My baby and my husband are Italian nationals. Can I travel with them with only the permesso di soggiorno? What could happen if I use 2 cycle fuel in a 4 cycle engine? First off, this is NOT to be confused with the following question:    2 cycle oil is used for my weed eater. There is no way I could use a whole gallon in a year.  Sometimes I run out of gasoline for my 4 cycle lawn mower, and that 2 cycle mix fuel is sitting there getting really old.  If I run that in the lawn mower, am I simply going to create extra pollution or could the 2 cycle fuel cause any damage to the 4 cycle engine? Word for people who live in the same city What are the people who live in the same city are called? Any words for that? I want to use it in the following context:      I and my ____ are happy. Does this email mean that I have been granted the visa? How can see call history in iPhone one month back, i also tried with tried party software but that is not perfect , is the any solution from apple? Is it possible to write any bounded continuous function as a uniform limit of smooth functions Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?   On any bounded interval this is true since by Stone-Weierstrass polynomials are smooth and dense in the continuous functions. I guess the result is still true for $\mathbb{R}$ but I don't know how to prove it. Since I don't have any integrability conditions on $f$ the technique of convoluting with mollifiers is probably not applicable. Rotating an object continously with OnMouseDown I have a script that works but it does not do what I want it to do. My script makes an object rotate on when another is clicked. However, I want the object to continuously rotate until I click on the other object again. Right now when I click it rotates just a bit and stops.  Here is my script:   public class Rotate : MonoBehaviour {     public float speed =50f;     public GameObject target;      public void OnMouseDown(){        target.transform.Rotate (new Vector3(0, Time.deltaTime, 0) * speed);     }  }   How can I make the object rotate continuously until I click it again? Installing Ubuntu on one of two hard drives The speed of light is the same in all inertial frames.  Does it change from a non-inertial frame to another? Can it be zero?  If it is not constant in non-inertial frames, is it still bounded from above? At most finitely many (Hamel) coordinate functionals are continuous - different proof Adding Maximum and Minimum Latitude and Longitude as features in ArcGIS for Desktop? I am a graduate student in economics who recently converted to R from other very well-known statistical packages (I was using SPSS mainly). My little problem at the moment is that I am the only R user in my class. My classmates use Stata and Gauss and one of my professors even said that R is perfect for engineering, but not for economics. He said that many packages are built by people who know a lot about programming, but not much about economics and therefore are not reliable. He also mentioned the fact that since no money is actually involved in building an R package, there is therefore no incentive to do it correctly (unlike in Stata for example) and that he used R for a time and got some "ridiculous" results in his attempts to estimate some stuff. Moreover, he complained about he random number generator in R which he said was "messy".  I've been using R for just a little more than a month and I must say I have fallen in love with it. All this stuff I am hearing from my professor is just discouraging me.  So my question is: "Is R reliable for the field of economics?". I've imported several shapefiles which where exported from a Mapinfo .tab. Several  .tab-files are imported resulting in 20+ layers. Afterwards I want to export it to GeoJSON; but I'm reluctant to select each layer and export it manually. Is there a way to merge all the layers into one using QGIS? Don't let the association bonus be given away as bounties When a user with at least 200 rep on any Stack Exchange site registers a new account on another SE site, they immediately receive 100 rep on that site as an .  This is a deliberate feature to give users who are already familiar with how the SE platform and community works a bit of an extra boost (e.g. letting them upvote and comment immediately), and it's pretty much working as intended. What's probably not intended is that, currently, this bonus rep can be given away as . In fact, because awarding bounties currently requires only 75 rep, any user with 200+ rep on at least one site can join another site, receive the 100 rep bonus, and immediately award it all as a bounty to another user, without having to earn any actual rep on the site at all.  Not only does this seem contrary to the design intent of bounties (and of the association bonus), but it also seems ripe for potential abuse. (In fact, the only thing stopping a sockpuppet ring from gaining unlimited rep this way is the fact that the bounties would likely be sooner or later noticed by mods.) It's also possible for experienced users to (more or less) accidentally bounty their reputation so low that that they lose access to basic site features that they'd normally have due to the bonus.  Arguably, of course, this is entirely the user's own fault, but it can still be .  Also, this behavior , which currently require 110 rep to answer for users with the association bonus. The fix I'd like to suggest is simple:  If a user has been awarded the association bonus, don't let them award a bounty that would take their rep below 100.  Of course, it would still be possible for users with the bonus to drop below 100 rep via other means, e.g. by having their posts downvoted.  But they wouldn't be able to just give away this baseline rep to other users any more. What is the difference between Halt and Shutdown commands? What resolution should a large format artwork for print be? For wall-sized graphics and large banners (e.g. 3m × 5m), what is an acceptable PPI for print? As I understand it 300 ppi is the typical for 'small' artworks (esp. for clean text resolution). However, for small artworks, the audience usually engages at a much closer distance. Therefore, in the interests of keeping a bitmap file down to a reasonable size, can you get away with a much-reduced PPI for a large format/banner print? Create longer arrows Circular polarizing filter - slim or regular? I have a Canon EOS 550D. Right now I only have some standard lenses, and I'm looking to buy a Tokina AT-X 116 Pro DX II 11-16mm F/2.8. (I love landscape photography and photography in nature in general.) I want to try using a circular polarizing filter, and I've been looking at the Hoya Filter Circ Pol DMC Pro1 77mm. I can see that the "slim" version is cheaper than the regular one, however, I'm not sure what the difference is. Can any of you tell me which would one be preferable for landscape photography? (I found a thread posted about this some years ago but I would like to hear what you guys think, knowing what camera, lens and filter I'm thinking about combining.)  All feedback would be very much appreciated! :-) If an polynomial has a complex root, is it necessary that the conjugate of that root is also a root of the polynomial? I am not sure about this. Please comment on this.