I need to prove that : $\mathrm{\sum_{n=0}^{\infty} {p^n\over n!} |p-n| = 2{p^p\over(p-1)!}}$ Knowing that $\mathrm{\sum_{n=0}^{\infty} {p^n\over n!} = exp(p)} $ I've tried many ways : $\mathrm{\sum_{n=0}^{\infty} |p-n| = \sum_{n=0}^{p-1} (p-n) + \sum_{n=p+1}^{\infty} (n-p) }$ $\mathrm{ \Longleftrightarrow ... \Longleftrightarrow \sum_{n=0}^{\infty} |p-n| = \sum_{n=0}^{p} n + \sum_{n=0}^{\infty} n }$ Also tried to use ln()... But nothing seems to be leading to a result... I have spent more than 3 hours just on this one... Can anyone show me the answer and the way I need to proceed in such exercises ?
I want to show that for each $n \in \mathbb N$ it holds that $$\sum_{k = 0}^\infty \frac{n^k}{k!} \vert k - n \vert = 2n \frac{n^n}{n!}.$$ But I have absolutely no idea how to do that. I tried to split the series like this:  $$\sum_{k = 0}^\infty \frac{n^k}{k!} \vert k - n \vert = \sum_{k = 0}^{n - 1} \frac{n^k}{k!}(n - k) + \sum_{k = n + 1}^\infty \frac{n^k}{k!} (k - n)$$ Now I want to show that $\sum_{k = 0}^{n - 1} \frac{n^k}{k!}(n - k) = \sum_{k = n + 1}^\infty \frac{n^k}{k!} (k - n) = n \frac{n^n}{n!}$ holds. But I got no idea how to achieve that. I tried to use induction but things became pretty messy and I don't think it will work like that. On the other hand I got the idea that it might have to do with $\exp$ or binomial coefficients but I don't see where I can facilitate that. I would appreciate some hints :)
How to identify polygons with "flag" sliver errors Working in ArcMap, I have come across errors in a polygon layer that I will call "flag slivers" (taken from similar language in parcels call "flag lots").  These sliver polygons (see image below) are typically composed of just one additional node that user has  accidentally added.    In the images below, the "flag" and the "flagpole" are one single-part feature where the "flagpole" just overlaps itself.  The left-hand image has a total of 5 nodes.     When seen alone the errors are very obvious, but when multiple polygons are adjacent, they are nearly impossible to see because they appear to be the boundary between 2 polygons.  This likely happens because they are using a shapefile based editor, and therefore I cannot implement any topology-based editing rules to prevent this from happening in the future.  Does anyone have a way of identifying and resolving these types of errors?  I would prefer an automated method of both identification and resolution since field users are the ones who created the errors, but I am stuck cleaning up after them.  Thanks. Prove that $4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$ . Prove that $$4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$$   I think midway calculations are not easy formulas, I can't find what kind of formula transformation to solve! I think the best solution here is using right triangle . . . I have one solution too, but not pretty . ! Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1? I was wondering why do some people use redundant axioms in definitions?  If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ab and 1. Then, you get it's commutative for free.   Why do we then assume it?   Also, is it even stronger. If we drop the condition that 1 is in R, then can we still deduce a+b=b+a? As clearly the argument just done falls apart if you haven't got 1 in R. On a quantum scale the smallest unit is the , which is a discrete measure.  There several question that come to mind:   Does that mean that particles can only live in a discrete grid-like structure,  i.e. have to "magically" jump from one pocket to the next? But where are they in between? Does that even give rise to the old paradox that movement as such is impossible (e.g. )? Does the same hold true  for time (i.e. that it is discrete) - with all the ensuing paradoxes? Mathematically does it mean that you have to use difference equations instead of differential equations? (And sums instead of integrals?) From the point of view of the space metric do you have to use a discrete metric (e.g. the ) instead of good old Pythagoras?   Thank you for giving me some answers and/or references where I can turn to.  Update: I just saw this call for papers - it seems to be quite a topic after all: Is Reality Digital or Analog? FQXi Essay Contest, 2011. , , . One can find some pretty amazing papers over there. Sequence of differentiable functions Let $f_n$ be a sequence of differentaible functions on $[0,1]$ to $\mathbb{R}$ converging uniformly to a function $f$ on $[0,1]$, Then    $f$ is differentiable and Riemann integrable there $f$ is uniformly continuous and R-integrable $f$ is continuous, need not be differentiable on $(0,1)$ and need not be R-integrable on $ [0,1]$, $f$ need not be continuous.   Well I think 2 is only correct statement  as every $f_n$ are differentiable on $[0,1]$ hence uniformly contnous, and as they converge uniformly to $f$ on $[0,1]$ so $f$ is uniformly continuous and every continuous function is rieman integrable..Am I correct? How do I restore iPhone 6s stuck on connect to iTunes screen/Apple logo I'm not really sure how, but after I woke up today, my phone was just displaying the loading spinner and doesn't start up anymore. It's weird because my phone turned off last night and suspecting that it was probably out of charge, I thought charging it overnight would fix the problem.   So, my iPhone is currently stuck on the iTunes/Apple Logo loop. When I put it into DFU/Recovery Mode and connect it to iTunes, it isn't able to restore it. It keeps throwing "Network Connection Problems" -- it downloads iOS 10 for a bit and suddenly stops downloading, throwing the error. I've also tried downloading a custom .ipsw from  but it seems that the file is corrupted whenever iTunes tries to unpack it.   Any thoughts on how I could go about fixing this?   Note - I don't have a jailbroken iPhone. Centos mysql version is 5.5 however PHPmyadmin still says version 5.1 When umpire change the cricket ball, was the old ball thrown as waste or keep for reuse? Getting key with maximum value in dictionary? Case expressions in Access How is double negation interpreted in English? In my native language, double negations "cancel out". I have been trying to learn English using movies and such things, but double negation confuses me. I see instances when double negation cancels out, and some when they do not.  For example, "I haven't got no money" is interpreted as "I have no money." "There wasn't anyone who didn't enjoy the party" is interpreted as "Everyone enjoyed the party."  Is this context dependent? If so, how does one decide which interpretation is correct? Can Conjure Animals beasts be used as mounts? Is it possible to use the animals that are summoned using Conjure Animals as mounts?  As an example: Summoning Giant Eagles to carry a group up the mountain or over a wall. Is there a simple way to convert C++ enum to string? Is there a definitive list of which episodes / scenes that they used for the classic doctors in The Day of The Doctor? Is there a single word to describe a solution that hasn't been optimized? "How are" or "How is" the wife and kid? How are the wife and kid?   How is the wife and kid?      Which is more correct? Geometric interpretation of Electromagnetism For gravity, we have    General Relativity, which is a geometric theory for gravitation.  Is there a similar analog for Electromagnetism? Erf as term for plot of land in a town or city. Is this word only used in southern Africa? The Free Dictionary website states and so does Wikipedia:     erf [ɜːf]   n pl erven [ˈɜːvən]   (Engineering / Civil Engineering) South African a plot of land, usually urban, marked off for building purposes   [Afrikaans]   The term erf is used in legal documentation and technical English throughout South Africa and Namibia, maybe even further afield in the region.  Is it used anywhere else, except maybe the Dutch/Flemish regions in Europe, where I assume the term originated?  An example:      The client purchased Erf 132, Windhoek for a large sum of money. He intends to build a house on this erf.   The capitalisation of "Erf 132" is commonplace, since it is a definition of a location. This is similar to capitalisation of towns or cities. I find it slightly awkward, but that is how it is done. I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. How to connect to wifi using just the terminal?