$\newcommand{\intd}{\,\mathrm{d}}$On an AOPS forum a user solved that  $$\int\frac{1}{x(x+1)(x+2)\ldots(x+n)}\intd{x}=\frac1{n!}\sum_{k=0}^n(-1)^k\binom{n}{k}\ln|x+k|+C$$ This to me is absolutely amazing and their solution interesting enough, I am curious as to how I would go about proving this. To me this shouts induction, it is obvious for $n=0$ ($\int\frac1x\intd{x}$) however on the inductive step I could not figure out what to do. Integration by parts did not help and beyond that I am at a loss. I look forward to your responses
I want to derive a closed formula for the integral $$I_n= \int \frac {dx}{\prod_{r=0}^n (x+r)}$$  On writing out first few terms we get   For $n=0$, $$I_0=\ln \vert x\vert+C$$ For $n=1$, $$I_1=\ln \vert x\vert-\ln \vert x+1\vert+C$$ For $n=2$ $$I_2=\frac {1}{2!}\left(\sum_{r=0}^2 (-1)^r\binom {2}{r} \ln \vert x+r\vert\right)+C$$ For $n=3$ $$I_3=\frac {1}{3!}\left(\sum_{r=0}^3 (-1)^r\binom {3}{r} \ln \vert x+r\vert\right)+C$$  Hence for generalized $n$ we have $$I_n=\frac {1}{n!}\left(\sum_{r=0}^n (-1)^r\binom {n}{r} (\ln \vert x+r\vert)\right)+C$$  Now this is just an observation but I want to prove that it is correct. I have tried lot of methods but not useful. Partial fractions would have been most useful but would go out on tedious task which is nearly impossible.  Also integration by parts won't help nor any trig substitution. So any ideas are welcome.  And ya,  this is not a homework question,  it's a question which I just saw in a integral challenge paper.
TTY doesn't wait for password I upgraded from 16.04 to 18.04. When I press Ctrl+Alt+F3 I get the terminal which prompts for the username. When I type my username and press Enter, it doesn't wait for my password (its as if I pressed Enter without entering anything). Even if I enter something, the text shown in plaintext and then it keeps on saying my password is wrong. After few times, the screen is cleared and it prompts for my username again and the same loop goes on. I saw many saying that the latest version of Ubuntu 16.04 includes an opensource driver for ATI. I have an HP laptop with AMD 6770m + Intel HD 3000 hybrid graphics and wants to install Kubuntu on it. I created a bootable USB drive of Kubuntu 16.04 and booted from it just to check everything works perfectly (I don't want to install it separately and ruin bootloader as of now). But my laptop is getting too hot and the battery is not lasting for more than 30 mins. Also the brightness control is not working at all and it stays high always.   How can I get hybrid graphics on Kubuntu/Ubuntu working well? I am ready to turn off the ATI card since I am not going to play any games on Linux. I just need the Intel graphics + Cool Laptop + Good battery Life + Brightness control.  My BIOS setting for switchable graphics mode is FIXED. Do I need to change it to DYNAMIC?  UPDATE  I wasn't able to make things work in 16.04 so installed 14.04 LTS along with the AMD Catalyst driver. Now everything works perfectly fine. Below are the steps.   Clean install Ubuntu 14.04 LTS Install the fglrx catalyst drivers using “Additional Drivers” window. This will enable us to switch graphics Restart Ubuntu FAN Noise Fix - Refer this  and install i8kutils Brightness Control - Check this     is the official documentation for AMD/Intel hybrid graphics Connect two computers with SSH in a home LAN I have two ubuntu computer shared the same wifi and desire to reach another from one's terminal.  First create an user on the second computer named "Second" and hostname "Algorithms"    Then tried to reach it from the other with SSH,  but get error report as   $ ssh second@Algorithms   ssh: Could not resolve hostname algorithms: nodename nor servname provided, or not known   The solution I tried is based on   What's the problem with my experiment? I have a directory filled with files with names like logXX where XX is a two-character, zero-padded, uppercase hex number such as:  log00 log01 log02 ... log0A log0B log0C ... log4E log4F log50 ...   Generally there will be fewer than say 20 or 30 files total.  The date and time on my particular system is not something that can be relied up on (an embedded system with no reliable NTP or GPS time sources).  However the filenames will reliably increment as shown above.  I wish to grep through all the files for the single most recent log entry of a certain type, I was hoping to cat the files together such as...  cat /tmp/logs/log* | grep 'WARNING 07 -' | tail -n1   However it occurred to me that different versions of bash or sh or zsh etc. might have different ideas about how the * is expanded.  The man bash page doesn't say whether or not the expansion of * would be a definitely ascending alphabetical list of matching filenames.  It does seem to be ascending every time I've tried it on all the systems I have available to me -- but is it DEFINED behaviour or just implementation specific?  In other words can I absolutely rely on cat /tmp/logs/log* to concatenate all my log files together in alphabetical order? House computer with AI takes over home, tries to procreate with woman Fitting linear model / ANOVA by group When doing an apt-get update/upgrade I see that warning  W: Conflicting distribution: https://downloads.plex.tv/repo/deb public InRelease (expected public but got )   I was upgrading to Ubuntu Zesty 17.04 before and do not know how to fix the above. Already have seen ? but there are no suggested fixes for that.  Maybe one of you can give some hints? Thanks! When it comes to getting a tenure-track position where one has both a lot of resources and academic freedom to do what one wants to do (so having a tenure-track position in a top university might be more useful because one may get more resources at a top university).   Or in short, when one wants to maximize one's chances of becoming a "top scientist".  There are obviously numerous factors, and "rank/prestige" should not be considered as a factor in itself. That said, the period after one gets a PhD is an extremely important time for building up connections, and those are easier to get at a top university. Understanding slice notation I need a good explanation (references are a plus) on Python's slice notation.   To me, this notation needs a bit of picking up.   It looks extremely powerful, but I haven't quite got my head around it. I'm having trouble understanding Cantor's diagonal argument.  Specifically, I do not understand how it proves that something is "uncountable".  My understanding of the argument is that it takes the following form (modified slightly from the  article, assuming base 2, where the numbers must be from the set $ \lbrace 0,1 \rbrace $):  \begin{align} s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In this case, the diagonal number is the bold diagonal numbers $(0, 1, 1)$, which when "flipped" is $(1,0,0)$, neither of which is $s_1$, $s_2$, or $s_3$.  My question, or misunderstanding, is: When there exists the possibility that more $s_n$ exist, as is the case in the example above, how does this "prove" anything?  For example:  \begin{align} s_0 &amp;= (1,0,0,\mathbf{0},\dots)\ \ \textrm{ (...the wikipedia flipped diagonal)}\\ s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ s_4 &amp;= (0,1,1,\mathbf{1},\dots)\\ s_4 &amp;= (1,0,0,\mathbf{1},\dots)\ \ \textrm{ (...alternate, flipped } s_4\textrm{)}\\ s_5 &amp;= (1,0,0,0,\dots)\\ s_6 &amp;= (1,0,0,1,\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In other words, as long as there is a $\dots \text{ continues}$ at the end, the very next number could be the "impossible diagonal number", with the caveat that it's not strictly identical to the "impossible diagonal number" as the wikipedia article defines it:     For each $m$ and $n$ let $s_{n,m}$ be the $m^{th}$ element of the $n^{th}$ sequence on the list; so for each $n$,      $$s_n = (s_{n,1}, s_{n,2}, s_{n,3}, s_{n,4}, \dots).$$      ...snip...      Otherwise, it would be possible by the above process to construct a sequence $s_0$ which would both be in $T$ (because it is a sequence of 0s and 1s which is by the definition of $T$ in $T$) and at the same time not in $T$ (because we can deliberately construct it not to be in the list). $T$, containing all such sequences, must contain $s_0$, which is just such a sequence. But since $s_0$ does not appear anywhere on the list, $T$ cannot contain $s_0$.      Therefore $T$ cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.   I'm not sure this definition is correct, because if we assume that $m = (1, \dots)$, then this definition says that "$s_n$ is equal to itself"&mdadsh;there is no "diagonalization" in this particular description of the argument, nor does it incorporate the "flipping" part of the argument, never mind the fact that we have very clearly constructed just such an impossible $T$ list above.  An attempt to correct the "diagonalization" and "flipping" problem:  $$s_n = (\lnot s_{m,m}, \lnot s_{m,m}, \dots) \quad \text{where $m$ is the element index and} \quad\begin{equation}\lnot s_{m,m} = \begin{cases}0 &amp; \mathrm{if\ } s_{m,m} = 1\\1 &amp; \mathrm{if\ } s_{m,m} = 0\end{cases}\end{equation}$$  This definition doesn't quite work either, as we immediately run in to problems with just $s_1 = (0),$ which is impossible because by definition $s_1$ must be $ = (1)$ if $s_1 = (0)$, which would also be impossible because...   Or more generally, with the revised definition there is a contradiction whenever $n = m$, which would seem to invalidate the revised formulation of the argument / proof.  Nothing about this argument / proof makes any sense to me, nor why it only applies to real numbers and makes them "uncountable". As near as I can tell it would seem to apply equal well to natural numbers, which are "countable".  What am I missing? In Days Of Future Past, Quicksilver spends a lot of time playing with electronics.  In his famous tour de force scene in the Pentagon, he specifically puts his headphones on before running around the kitchen.  But why?  Does the music on his Walkman speed up with him as he uses it, or is it still operating at "normal" speed? Or indeed, does any electronic device that he touches speed up? Query to find GUID's "not in" null containing subquery returns no results What is the difference between private and protected members of C++ classes? What is the difference between private and protected members in C++ classes? I understand from best practice conventions that variables and functions which are not called outside the class should be made private—but looking at my MFC project, MFC seems to favor protected. What's the difference and which should I use? I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?  If we can't get a free quark then what is single-top-quark? Young boy living on Mars discovers that his parents are robots On the meta home page I noticed the  is listed as doing something, but cannot figure out what it is.  There is nothing on the  or its , but the recent activity is listed with the same timestamp. The full question: suppose N is normal in G and $[G:N]=p$ is a prime. Prove that for any $K \leq G$, either (1) $K \leq N$ or (2)$KN=G$ and $[K:K \cap N]=p$.  Since $N,K \leq G$, it is of course possible that $K \leq N$ could happen.  but for (2), we can show $KN \subseteq G$  but how can we show $KN=G$?  And what information does "$[G:N]=p$" give us? Automatically protect questions with three or more low-rep-user deleted answers There are a few "automatically protect questions" requests out there, but I'd like to add another twist.  I'd like to request that questions that have three or more deleted answers by users with &lt;50 rep be automatically protected.  The reasoning behind this is that we have lots and lots of "not an answer" flags in the mod queue.  But we cannot protect a question from the mod queue, so we have to open each answer, evaluate if the question needs protecting, then protect it.    This process takes a long time, and can definitely be automated, as the only thing you really need to check is how many new users are posting non-answers or are spamming their awful products. Given a template where the HTML cannot be modified because of other requirements, how is it possible to display (rearrange) a div above another div when they are not in that order in the HTML? Both divs contain data that varies in height and width.  &lt;div id="wrapper"&gt;     &lt;div id="firstDiv"&gt;         Content to be below in this situation     &lt;/div&gt;     &lt;div id="secondDiv"&gt;         Content to be above in this situation     &lt;/div&gt; &lt;/div&gt; Other elements   Hopefully it is obvious that the desired result is:  Content to be above in this situation Content to be below in this situation Other elements   When the dimensions are fixed it easy to position them where needed, but I need some ideas for when the content is variable. For the sake of this scenario, please just consider the width to be 100% on both.  I am specifically looking for a CSS-only solution (and it will probably have to be met with other solutions if that doesn't pan out).   There are other elements following this. A good suggestion was mentioned given the limited scenario I demonstrated&mdash;given that it might be the best answer, but I am looking to also make sure elements following this aren't impacted. Why can't two series-connected diodes act as a BJT?