What is the sum of the given special series?any specific formula? $1^n$+$2^n$+$3^n$+$\dots$+$m^n$. $(m,n\in\mathcal N)$  $m,n\ge2$
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer.  In an attempt to make more interesting answers, I ask that this question concern the problem of "Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula for fixed $p\in\mathbb N$".  I've even checked this  without finding what I want.  Rule #1: Any method to compute the sum in question for arbitrary $p$ is good, either recursively or in some manner that is not in itself a closed form solution.  Even algorithms will suffice.  Rule #2: I don't want answers confined to "only some values of $p$".  (A good challenge I have on the side is a generalized geometric proof, as that I have not yet seen)  Exception: If your answer does not generalize to arbitrary $p$, but it still generalizes to an infinite amount of special $p$'s, that is acceptable.  Preferably, the method is to be easily applied, unique, and interesting.  To start us off, I have given my answer below and I hope you all enjoy.
How to get an enum value from a string value in Java? Say I have an enum which is just public enum Blah {     A, B, C, D }  and I would like to find the enum value of a string, for example &quot;A&quot; which would be Blah.A. How would it be possible to do this? Is the  the method I need? If so, how would I use this? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. How to edit / how to format box overlapping with footer when scrolling down Why is $c+q$ still a primitive root modulo $q$? Question: Let $p$ and $q$ be distinct odd prime numbers.  By considering primitive roots, we need to show $\exists c\in\mathbb{Z}$ with the property that:  $\bullet$ $c^n\equiv 1\pmod{p}$ whenever $n$ is a multiple of $p-1$, and   $\bullet$ whenever $n$ is not divisible by $q-1$, then $c^n\not\equiv 1\pmod{q}$  I am only interested with the beginning of this answer.  Answer: We let $c$ be a primitive root modulo $q$.  We are going to use Fermat's little theorem so we need $(c,p)=1$  If $(c,p)\neq 1$, then we have $p\mid c$, but then we have $(c+q, p)=1$  and since $c+q$ is a primitive root modulo $q$, we can replace $c$ with $c+q$ for $(c,p)=1$ for using Fermat's little theorem.  I understand the general method and argument, but...  Contention: Why is $c+q$ still a primitive root modulo $q$?  When is this generally true? Is there an identity for when things remain a primitive root?  Why is $(c+q)^{\phi(q)}\equiv 1\pmod{q}$? I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that     Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance. ()   But if this is the case how does a point-like particle know, which direction it should follow in a gravitational field?   Can this be a seen as a hint that point-like particles cannot be used as underlying concept for a field theory like general relativity? convert bash `ls` output to json array Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that   $$\int_{0}^\infty|f(t)|dt&lt;\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$  If so can you provide a proof, otherwise a counter example. Thank you. How much effect does the Bernoulli effect have on lift? I understand that the Bernoulli effect is a flawed explanation for the cause of lift, and does not cause much at all, but how much?   Is there any experimental data on the force caused by the Bernoulli effect? Maybe implicitly through data of the pressure difference between the top and underside of an aeroplane's wings. After that, I assume I could (crudely approximating the pressure to be acting perpendicularly to the flight direction) use $\Delta P A$ to work out the net force on the plane.    Perhaps there is another way to quantitatively analyse the extent to which the Bernoulli effect causes lift.  Edit: see  short cartoon (content similar to Mike Dunlavey's answer). how many instances for a class exists at runtime in java How do I know whether to use the definite (the) or indefinite (a, an) article, and when to omit it altogether? limit of $\lim\limits_{x \to \infty}((x+1)^a - x^a)$ Hello everybody, i couldn't figure out how to demonstrate this equation without L'Hopital i wonder if its possible.   $a \in (0,1)$ $\lim\limits_{x \to \infty}((x+1)^a - x^a) = 0$ How to take mysqldump with Generated column? I am getting below error while importing the dump,     ERROR 3105 (HY000) at line 82: The value specified for generated   column 'column_name' in table 'table_name' is not allowed.   When I describe the table with error, there are some generated columns in it.     option_type         | varchar(45)  | YES  | MUL | NULL    | VIRTUAL   GENERATED   What is the correct command to use when taking mysql dump for the database contains generated column?  MySQL server source and destination version:     mysqld Ver 5.7.26 for Linux on x86_64 (MySQL Community Server (GPL))   MariaDB mysqldump CLI version:     mysqldump Ver 10.17 Distrib 10.3.12-MariaDB, for Linux (x86_64) Side-by-side markdown title not so side-by-side In edit revisions where a long enough title has been edited, the side-by-side  markdown view shows a large overlap when the title wraps:    This is a repeat of  specific to the new Meta design. Looks like a line spacing adjustment is needed.  See this . There are several spells that require the target to be "humanoid" such as , , etc.  However, if the target is Polymorphed after that spell has been successful, then what happens?  Does the change in form to Beast end the spell effects? Does it suppress the spell until the correct form returns? Does it do something else? Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$ Where are Upstart log messages on Ubuntu 13.X? I'm looking for a simple, easy to understand algorithm to alphabetically sort an array of characters in C. Basic Excel currency format with Apache POI I'm able to get cells to format as Dates, but I've been unable to get cells to format as currency... Anyone have an example of how to create a style to get this to work?  My code below show the styles I'm creating... the styleDateFormat works like a champ while styleCurrencyFormat has no affect on the cell.  private HSSFWorkbook wb; private HSSFCellStyle styleDateFormat = null; private HSSFCellStyle styleCurrencyFormat = null;   ......  public CouponicsReportBean(){     wb = new HSSFWorkbook();     InitializeFonts();  }  public void InitializeFonts() {     styleDateFormat = wb.createCellStyle();     styleDateFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("m/d/yy"));       styleCurrencyFormat = wb.createCellStyle();     styleCurrencyFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("$#,##0.00"));  } I have a question related to multiple regression and interaction, inspired by this CV thread:    When checking for a moderation effect I do center my independent variables and multiply the centered variables in order to calculate my interaction term. Then I run my regression analysis and check for main and interaction effects, which may show the moderation.  If I redo the analysis without centering, apparently the coefficient of determination ($R^2$) does not change but the regression coefficients ($\beta$s) do. That seems clear and logical.  What I do not understand: The p-values of the main effects change substantially with centering, although the interaction does not (which is right). So my interpretation of main effects could change dramatically - just determined by centering or not. (It's is still the same data, in both analyses!)  Can somebody clarify? - Because that would mean that the option to center my variables would be mandatory and everybody should do it in order to get the same results with the same data.    Thanks a lot for distributing to that problem and your comprehensive explanations. Be assured that your help is very much appreciated!  For me, the biggest advantage of centering is to avoid multicollinearity. It is still quite confusing to establish a rule, whether to center or not. My impression is, that most of the resources suggest to center, although there are some "risks" when doing it. Again I want to put out the fact, that 2 researchers dealing with the same material and data might conclude different results, because one does centering and the other does not. I just read some part of a book by Bortz (he was a Professor and kind of a Statistics Star in Germany and Europe), and he does not even mention that technique; just points out to be careful in interpreting main effects of variables when they are involved in interactions.  After all, when you conduct a regression with one IV, one moderator (or second IV) and a DV, would you recommend to center or not? Google search results showing up a URL on its search which doesn't exist on the website I am using a keyword to display a website page on google search. After entering the keyword, on the first page of google, I am redirected to a page, whose domain is exactly similar to the website whose page I want to display, but it is showing a page which doesn't even exist on the domain. Also, as similar domain with spam characteristics is also getting displayed.   To practically demonstrate this issue, do follow the below steps.   Go to  Type dolce streichquartett and click on search. The 4th/5th URL has this link http://www.dolce-streichquartett.de/?high_ry=/casinos/de/spielautomaten-casino-bregenz.php, which is no where on the website and looks like spam The 7th/8th link you see is http://deonlinecasino.org/doc/kostenlos-roulette-spielen-spielgeld-ausdrucken.php, which is exactly similar to the above with the title as the keyword but a different domain.   What is this issue and why is this happening?