Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdots(n+k)$  e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + \cdots + n(n+1)(n+2)$  How does it do it?   EDIT: We can rewrite as: $\sum {(n+k)! \over n!} = \sum k!{(n+k)!\over n!k!} = \sum k!{{n+k} \choose n}$ (Thanks Daniel Fischer)  EDIT2: We can also multiply out and split sums. So e.g.  $$\sum (n-1)n(n+1) = \sum (n^3-n) = \sum n^3 - \sum n$$  But sums of powers actually seem to be more nasty than the original question, involving Bernoulli numbers. (Thanks Claude Leibovici)  And is there any name for this particular corner of maths? (i.e. How might I go about searching the Internet for information regarding this?)  PS please could we have a 'telescoping' tag?
Considering the following formulae:  (i) $1+2+3+..+n = n(n+1)/2$  (ii) $1\cdot2+2\cdot3+3\cdot4+...+n(n+1) = n(n+1)(n+2)/3$  (iii) $1\cdot2\cdot3+2\cdot3\cdot4+...+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4$  Find and prove a 'closed formula' for the sum  $1\cdot2\cdot3\cdot...\cdot k + 2\cdot3\cdot4\cdot...\cdot(k+1) + ... + n(n+1)(n+2)\cdot...\cdot (k+n-1)$  generalizing the formulae above.  I have attempted to 'put' the first 3 formulae together but I am getting nowhere and wondered where to even start to finding a closed formula.
Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$ Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q&lt;\infty} L^{q}$$ So,$\lim _{q \rightarrow \infty}\|f\|_{q}$ , $\|f\|_{\infty}$ are equal-norm with space $ L^{\infty} \cap\left(\bigcup_{p} \bigcap_{p \leqslant q} L^{q}\right)$. Case 1: $m(X)&lt;\infty $.It's easy to prove that. Case 2: $m(X)=\infty $. I have no idea about it,And I started to doubt the correctness of this conclusion. Can somebody give me a hint for this problem or just give an example to prove that this is a wrong conclusion when $m(X)=\infty $. Thanks in advance. I'm reading Linear Algebra by Hoffman, Kunze where the authors explained that a $n\times n$ matrix $A$ being invertible is equivalent to the fact that $A$ is row-equivalent to $n\times n$ matrix $R$ which is an identity matrix.  In the proof of the theorem, they wrote:     $$R= E_k\ldots E_2E_1 A$$ where $E_1,\ldots,E_k$ are elementary matrices. Each $E_j$ is invertible, and so $$A = E_1^{-1}\ldots E_k^{-1}~R\,. $$  ... Since, $R$ is a (square) row-reduced echelon matrix, $R$ is invertible if and only if $R=I\,.$  [...]   I couldn't get the conclusion, since any row of $R$ can't be zero, it has to be identity matrix $I\,.$ Why is it so?  Isn't there any other row-reduced echelon matrix other than the identity matrix having no zero row and invertible? Why is it so? What have to be normally distributed: groups or whole sample? It is a general and maybe basic question, but it will help me to avoid a mistake. To test the dependence of continuous variable on nominal with parametric test (t-test, ANOVA), "the data has to be normally distributed" (distribution have to be not much violated from normal).   But I don't clearly understand: I have to test the normality of the continuous data together (whole sample) or the normality of data in every single group?   And if my groups have small sizes (n=4-6), and therefore I fail to reject null, does this indicate a permission to use parametric methods?  Can anyone provide me a link to some good examples of statistical (biomedical) hypothesis testing with R (google did not helped)? Is there any way to do n-level nested loops in Java? In other words, can I do something like  for() {     for {        for {        }     } }   Except N times? In other words, when the method creating the loops is called, it is given some parameter N, and the method would then create N of these loops nested one in another?  Of course, the idea is that there should be an "easy" or "the usual" way of doing it. I already have an idea for a very complicated one. How do I change JPanel inside a JFrame on the fly? To put it simple, there's a simple java swing app that consists of JFrame with some components in it. One of the components is a JPanel that is meant to be replaced by another JPanel on user action.  So, what's the correct way of doing such a thing? I've tried  panel = new CustomJPanelWithComponentsOnIt(); parentFrameJPanelBelongsTo.pack();   but this won't work. What would you suggest? Simple & easy way to jail users I need a simple and easy way to jail users in their home directories in Oneiric. Do you have a simple configuration for jailing users, with full help, or some good web links?  I would be offering an online free public server with 10 to 20 GB free space. I don't know how many users. I want to give them SSH and SFTP so that they can connect through FileZilla. Spurious space above align environment at top of page/minipage It seems that the align environment adds additional vertical space if it starts at the top of a page or minipage.  Below are two pages, and two minipages (on Page 1), where the first page/minipage has text followed by an align* environment, and the second page/minipage starts with an align* environment, and note that the equations are still aligned.  Attempted Solution: The closest related question I found was:    I attempted to use the  at the above by adding \setlength\abovedisplayskip{0pt} (after the preamble as per ).  Although, this did move the equations up vertically, they were still vertically aligned:  And since the problem occurs at top of a page as well, I don't think this is related to  Question:  How can I adjust the align environment to automagically not have additional space at the top of a page or minipage? Is there a reason for this behavior, and perhaps should not be changed?  Code: \documentclass{article} \usepackage[showframe,paperwidth=3.5in,paperheight=2.2in,margin=0.5in]{geometry} \usepackage{amsmath}  \begin{document}%\setlength\abovedisplayskip{0pt} Top of Page 1 \begin{align*}% Text above at top of page  e &amp;= m c^2 \end{align*}  \hrule \begin{minipage}[t]{0.45\linewidth}     Top of minipage 1     \begin{align*}% Text above this minipage      e &amp;= m c^2     \end{align*} \end{minipage} \begin{minipage}[t]{0.45\linewidth}     \begin{align*}% No text above in this minipage      e &amp;= m c^2     \end{align*} \end{minipage}  \newpage \begin{align*}% No text above at top of page  e &amp;= m c^2 \end{align*} \end{document} It would be very useful to be able to overload the . operator in C++ and return a reference to an object.  You can overload operator-&gt; and operator* but not operator.  Is there a technical reason for this? selectively make contact groups silent Rounding to nearest 100 First number needs to be rounded to nearest second number. There are many ways of doing this, but whats the best and shortest algorithm? Anyone up for a challenge :-)   1244->1200  1254->1300 123->100 178->200 1576->1600 1449->1400 123456->123500 654321->654300 23->00 83->100 How to match * with hidden files inside a directory How to match the hidden files inside the given directories   for example   If I give the below command it's not giving the result of the hidden files,     du -b maybehere*/*   how to achieve this simple using a single command instead of using   du -b maybehere*/.* maybehere*/*   as I need to type maybehere twice. Can you use class features while Polymorphed? More specifically I'm trying to understand if a Barbarian who has been polymorphed, either willingly or not would be able to . However this also applies to things such as , or , anything a player might want to continue to use while no longer themselves.  Many questions specify you're able to rage while Wild Shaped, as  specifies      You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.   However,  specifies the following,     The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains it's alignment and personality.   And     The creature is limited in the actions it can perform by the nature of its new form.   Unlike Wild Shape, nowhere does polymorph specify that you retain benefits from your class or race. Does this imply that class abilities do not carry over to polymorphed individuals? Where can I find a standalone installer for Windows 7 Service Pack 1? Can I apply for and receive a US visa when residing in country other than my own? I'm currently residing in a country other than my own, and am planning a business trip to the USA for a week. Being a Bulgarian citizen, I do not qualify for the ESTA visa waiver program, unlike other EU countries.  The US embassy in Sofia lists a  which they inexplicably describe as "really quite easy". Unfortunately, some of the steps involved would be a major inconvenience for me, as I'm not residing in the country currently, particularly the interview and the visa/passport delivery.  I was hoping it could be possible to apply for a visa from the country of current residence rather than my own, but I wasn't able to find any information on the US embassy's webpages concerning this. Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$ How to use the Bradhley Hand font? I've checked the  that can be used through beamer class. I've successfully used most of the fonts, but when I tried to include the Bradhley Hand font, it did not work (it shows ordinary characters)?! How can I use it?  For example, when I use the following :  \documentclass[serif]{beamer}  \usepackage{mathpazo}  \renewcommand{\rmdefault}{ibh}  \begin{document} \begin{frame} $\frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} -1) } } { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} } $ \end{frame} \end{document}   it gives: How can a module detect when the "access denied" page is output? How is it possible for a module to detect when Drupal is outputting the "access denied" page? I know how to do it with Drupal 6; I need to know how to do it with Drupal 7. Vote too old to be changed, unless post is edited Anyone seen this one yet? Vote too old to be changed, unless post is edited My vote was only a few hours old. Then I realized the answer was wrong. Updated my question and went to remove the vote. Even after updating question I still have the issue. Small issue, but frustrating.  See also: Is the set of all functions $f:Z_+\rightarrow Z_+$ that are eventually constant countable. Split seeds.rb into multiple sections?