I'm trying to prove  $$ \sum_{i=1}^n i.i! = (n+1)!-1 $$ with mathematical induction. The first step I did after prove it for 1 was: $$ \sum_{i=1}^{n+1} i.i! = (n+2)!-1= (n+2).(n+1)!-1 $$ but I can't do anything more.
Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially)   It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone arranged to height from shortest to highest, but it doesn't hold up for the LHS.   Alternatively, trying to visualize the LHS, I noticed that it's like a right angle tetrahedra:  1  2!+2!  3!+3!+3!  ...  But it doesn't help to see a connection to the RHS.  Note: no integrals or gamma function nor use of other identities without proving them nor generating functions.
Which programs can I use to edit PDF files? I know that there is a very similar question, but I'm not interested in highlighting. What I'm looking for is some tool in Linux to edit pdf files. I've tried pdfedit which did the work for me, but if freezes after a few edits. I've tried this tool under Ubuntu 9.10 and under KDE Fedora 14 and the same thing happens. I've tried Okular but the notations are too big to fill out exercises from pdf workbooks.    Is there any tool that suits this job? I don't care whether it is open source or I have to pay a reasonable price for it. Integrate $\int_{0}^{2\pi}\log|1-e^{i\theta}| d\theta$ Here is exercise 11, chapter 3 from Stein &amp; Shakarchi's Complex Analysis II:  Show that if $|a|&lt;1$, then:  $$\int_{0}^{2\pi}\log|1-ae^{i\theta}|\,d\theta = 0$$  Then, prove that the above result remains true if we assume only that $|a|\leq 1.$ ($a\in \mathbb{C}$)  I've already proved it for $|a|&lt;1$, but couldn't make it for $|a|=1$, which I thought would be the easy part. But the only thing I could come up with was that, WLOG, it is enough to prove that:  $$\int_{0}^{2\pi}\log|1-e^{i\theta}|\,d\theta = 0$$  Tried to look for some kind of symmetry, but could't find it. Any ideas? Thanks! Mean Value theorem problem?(inequality) I'm trying to solve this mean value theorem problem but confused where to start,     If $0&lt;a&lt;b$ prove that $(1-\frac{a}{b})&lt;\ln\frac{b}{a}&lt;\frac{b}{a}-1$   Can someone please lend me a hand? Why did the first Linux developers choose to implement a non-preemptive kernel? Is it to save synchronization?  As far as I know, Linux was developed in the early '90s, when PCs had a single processor. What advantage does a non-preemptive kernel give in such PCs? Why, however, the advantage is reduced by multi-core processors? "It's time something was / were done about the problem." It's time something was done about the problem.   It is a sentence from Murphy's "Grammar". Using was confused me, for me it should be were. It looks like this clause is in subjunctive mood&mdash;something that is unreal / imaginary, and in that case you always use were for any object.  Unless, of course, it is not the subjunctive mood, what would knock me down. How to disable windows snap target In a fresh install of Windows 10 Pro. When I drag a window a snap target position image is displayed. See image.    This is very frustrating as the image is often displayed under the place where I want to place the window. The affect being that the window is therefore snapped and not placed where I intended it to be.  How do I disable this?  To be clear I don't want to disable snapping altogether just the annoying destination image underlay  UPDATE  I do not have the option to "Allow the system to suggest companion windows when using Snap" Multiple TikZ keys with foreach Target='_blank' to show in new window, NOT new tab, possible? How can you programmatically tell an HTML SELECT to drop down (for example, due to mouseover)? How can you programmatically tell an HTML select to drop down (for example, due to mouseover)? Command line command to auto-kill a command after a certain amount of time I'd like to automatically kill a command after a certain amount of time.  I have in mind an interface like this:  % constrain 300 ./foo args   Which would run "./foo" with "args" but automatically kill it if it's still running after 5 minutes.  It might be useful to generalize the idea to other constraints, such as autokilling a process if it uses too much memory.  Are there any existing tools that do that, or has anyone written such a thing?  ADDED: Jonathan's solution is precisely what I had in mind and it works like a charm on linux, but I can't get it to work on Mac OSX.  I got rid of the SIGRTMIN which lets it compile fine, but the signal just doesn't get sent to the child process.  Anyone know how to make this work on Mac?  [Added: Note that an update is available from Jonathan that works on Mac and elsewhere.] Convert UTC date time to local date time In several cases I want to add a toolbar to the top of the iPhone keyboard (as in iPhone Safari when you're navigating form elements, for example).    Currently I am specifying the toolbar's rectangle with constants but because other elements of the interface are in flux - toolbars and nav bars at the top of the screen - every time we make a minor interface change, the toolbar goes out of alignment.  Is there a way to programmatically determine the position of the keyboard in relation to the current view? From what items can a wizard learn his spells? I'm now wondering after googling for a while, what are the different sources a Wizard can learn his spells from? Is it only limited to scrolls and tomes, or can a Wizard learn spells from other magic items?  The following quotes are from the PHB, page 114 Your spellbook      You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.      Copying a Spell into the Book   When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.      For each level of the spell, the process takes 2 hours and costs 50gp. The costs represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.   The way this is worded and considering the few examples we're given, shouldn't it be possible to learn a spell from more than just written sources?  An example would be Wand of Magic Missile: would it be possible for a Wizard to learn the spell Magic Missile by studying the spell when used by the item and write down his research and thus learning how to use the spell on his own and eventually master it? When can we speak of collinearity Should I do anything when I see several references that are not used in the text? In a linguistics paper I read, I noticed that several references in the bibliography are not actually used in the text. This is for about 10-15% of the references the case. Should I do anything with this? I always thought that all references must be used in the text, because even if they are just suggestions for further reading you would want some explanation why it is relevant and what you can find where.   I only know the author through their work and do not know the editors. The paper is from 2011. The references are rather general. They are relevant to the paper, but you would want page numbers with them unless you are familiar with them. They are all from different authors and different institutes. It is likely that the references were used in previous versions of the text. How to create an animated Electric Arc or Lightning? I know about the technique of using a displace modifier with an animated displace object on a thin plane, but there are no "branches", no forking in the lightning.  Here is an image  that shows more or less what I'm thinking of:    Is it possible to achieve an 3D animated affect like this?  I have tried a few things, such as the . However, it doesn't support animation and repeatedly generating meshes and then manually animating the visibility is a tedious. Example of a ring without Invariant Basis Property Pluralizing Noun Phrase? It is well-known that empty text files have zero bytes:    However, each of them contains , which according to my research, is stored in , and do use space.   Given this, it seems logical to me that it is possible to fill a disk by purely creating empty text files. Is this correct? If so, how many empty text files would I need to fill in a disk of, say, 1GB?    To do some checks, I run df -i but this apparently shows the % of inodes being used(?) rather than how much they weigh.  Filesystem             Inodes  IUsed    IFree IUse% Mounted on udev                   947470    556   946914    1% /dev tmpfs                  952593    805   951788    1% /run /dev/sda2            28786688 667980 28118708    3% / tmpfs                  952593     25   952568    1% /dev/shm tmpfs                  952593      5   952588    1% /run/lock tmpfs                  952593     16   952577    1% /sys/fs/cgroup /dev/sda1                   0      0        0     - /boot/efi tmpfs                  952593     25   952568    1% /run/user/1000 /home/lucho/.Private 28786688 667980 28118708    3% /home/lucho MacBook Pro locked with Find My Mac and won't let me boot