For which values $\alpha, \ \beta \in \mathbb R $ does the series $\sum_{n=2}^{\infty} \frac{1}{n^\alpha \ln^{\beta}(n)}$ converges?
For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ converge?
Limit of $n/\ln(n)$ without L'Hôpital's rule I am trying to calculate the following limit without L'Hôpital's rule:     $$\lim_{n \to \infty} \dfrac{n}{\ln(n)}$$   I tried every trick I know but nothing works. You don't have to prove it by definition. What is the purpose of a serial number? Which of the following is a correct sentence:   Does anyone know where's Linda?  Does anyone know where Linda is?    Does anyone know why and how to use above usage if #2 is correct. Also please let me know this kind of specific structures. Change email login to another id My original email id got compromised and now I want my new email id to be the main id for all SE sites . How to delete original id from the system as someone else might login from it ? Usually, in Star Trek, when there's a timeline change, the indication is that the original timeline is gone and replaced with the new one.  In City on the Edge of Forever, Kirk makes a comment that if they don't get the timeline straight, that millions of people who never died before will die.  Much of the time travel in Trek created suspense by declaring a need to restore the original timeline.  It's already been discussed in other questions about how, in the new movie (simply titled Star Trek with no subtitle or series number) they were unable to restore the original timeline.  While some of this is open to interpretation, it seemed like in this instance, for the first time, in Trek, there was an alternate timeline that was seen as an alternate possibility - no reference was made to it replacing the original timeline.  In this movie, was there any reference to the original timeline still existing?  In this movie, was the original timeline still considered to be intact? The spectral radius of normal operator $\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$. If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.  I started this way: $x^a=y^b$ $=&gt;$ $x=y^{b/a}=(y^{1/a})^b$. Thus $x$ is an integer, and so is $y^{1/a}$, so $y=n^a$ for some integer $n$ greater than $1$. Then from the given relation, we get $x=n^b$.   I am not very convinced with my solution, as I feel I did some mistake. Can anyone point out where the mistake is? Any other solution is also welcome :) Double dispatch in C#? Where is the line drawn with domain names which include a trademark? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks "I think him to be about 50" or "I think he is about 50"? How to add a gif file to my LaTeX file? Related:          I have an image file -- a gif file, which is the graph of a function. How do I add it to a LaTeX file? Will it come out OK in the subsequent pdf file? How does the new Tie-Breaker work? In Clash of Clans there was an update on 9/16 adding a Tie-breaker into the wars, can anybody explain in detail how this mechanism works? Is there a modal command for MakeLowercase? Analog to  question I've been wondering WHAT is to \lowercase as \bfseries is to \textbf? (... and for uppercase, or \MakeLowercase and MakeUppercase, etc., which also seem to be text-block commands as explained .)  Is there a "switch"/"declaration"/"modal command" (what is the correct term here?, is there a difference between them?) that would allow one to turn something into lowercase/uppercase in the same way that one would use \scshape for example, i.e., not as text-block command with the curly braces?  Or how to create such a command?  I couldn't find any clue on the forum. How to Backup and Restore Android OS without Rooting I want to find a way to restore my phone in case the process of rooting my phone or dual booting a custom rom breaks it. However, most of the ones I've looked at require the phone to be rooted before installing. I know I can create a nandroid backup through system settings, but I don't know how to recover that without rooting my device, or if there are better options. I'm not terribly concerned for any personal data on my phone (I can backup my pictures pretty easily), but I mostly just want to be able reinstall Android OS or reset to factory default in case my device breaks. Thanks! Edit: I am going to ask a new question (one different enough to justify not simply modifying this one) with the new info I learned from the possible duplicate. I thought of this question right after I posted a tweet about a service upgrading me to a free student account since I am in college. I said "That really helps a broke college student out." I actually paused for a second while I was writing that to decide if I should say     That really helps out a broke college student.   or     That really helps a broke college student out.   Are there any prescriptive rules about splitting phrasal verbs like this? I know this breaks the "don't put a preposition at the end of a sentence" rule, but that "rule" has an exception for phrasal verbs.  To clarify:   Is there any rule that says phrasal verbs can't be split, even if it is just an imposed, prescriptive rule? Does splitting a phrasal verb to put a preposition at the end of a sentence fall under the phrasal verb exception to the prepositions at the end of a sentence rule? Why isn't the symmetric twin paradox a paradox? Two twin sisters synchronize their watches and simultaneously (from the earth frame) depart earth in different directions. Following a predetermined flight plan, each sister accelerates identically to 99.9%c and then returns home at the same time (again in the earth frame). The observers on earth see each twin as having aged identically, as the symmetry of the problem dictates. Each twin however should see the other as having aged... What gives? Systems of equations with a single number Is just after your PhD a good time to switch your research area? I am finishing my PhD soon. The area in which I pursued my PhD is very challenging as I was only able to publish couple of papers even after spending many years. I want to switch into other research are due these reasons   to attract more students  to be comfortable to do research (along with teaching, I should be able to do research)   Even though I like the research I am doing right now, I seem to be isolated in my university. If ten students join my university for a PhD, hardly one student takes this area. My research area is not like machine learning or deep learning etc, so researchers of my domain are also not that much interested in my work.   The thing is I can work different research area but: Is this the right time? I am heading to a post-doc and learning a new area will probably take several months. Although my switch is not 180 degree, I am still worried. Is there a list of error codes for Minecraft? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. In the Iron Man movie Tony flies the Iron Man armour from Miami to Afghanistan, roughly 7800 miles.    What is the maximum range of the Iron Man armour?   What sort of speed does the armour cruise at?  How long would the flight from Miami to Afghanistan have taken?   Could Iron Man circumnavigate the globe without stopping, and how long might this journey take him?   EDIT: Just found this (funny) image, New York to Washington in under an hour?!? :)