I have been trying to solve such sums for a while involving limits of summations but haven't got any luck on this one yet...    $$\lim_{n\to \infty} \sum_{r=0}^n (r/n)^n$$.  I first thought of finding a series greater than this one and hoping that would be zero, so that I can claim that the given question is also zero. I could have used integration if there was another n in the denominator. And possibly, I don't think there is any way to sum up the numerator.
I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me?
What are the best ways to improve a site's position in Google? What are the best ways to improve a site's position in the Google search results? Is there any uncountably infinite set that does not generate the reals? Does there exist an uncountably infinite set $X \subseteq \mathbb R$ such that $\mathbb R \neq \left&lt;X\right&gt;$? I can't think of any, but I'm also having trouble trying to prove that no such subset exists.  For example: $\mathbb R$ is uncountable and obviously $\mathbb R = \left&lt;\mathbb R\right&gt;$. The Cantor set $C$ is uncountable, and we know that $C - C = [0, 1]$, so then since $\mathbb R = \left&lt;[0, 1]\right&gt;$ we know that $C$ also generates $\mathbb R$. Also the set of irrationals $\mathbb R \setminus \mathbb Q$ is uncountable, but we can generate all the rational numbers by fixing one irrational number $\alpha$ and then saying the any rational number $x$ shall be $(\alpha + x) - \alpha$, since both $\alpha + x$ and $\alpha$ are irrational.  So the examples that quickly come to mind all generate the reals. Is there a simple counterexample? Problems with # inside a verbatim that's inside a note I'm trying to use a # character inside a verbatim that's inside a note :   \begin{frame}[fragile]  \note{\begin{verbatim}  # \end{verbatim} }  \end{frame}   This produces the following errror :   >   ! Illegal parameter number in definition of \beamer@todo. >    to be read again>  >                     >   l.5 }   How can I write the # character in a way that doesn't break the latex compilation ?  Thanks in advanced,  PS : This is related to the following problem in pandoc : Why is integration the inverse of differentiation What circumstances can cause a question or answer to be deleted, and what does that actually mean?    How can a post be deleted? When can't I delete my own post? Can I still see my post even after it's deleted? Can I see a list of my deleted posts? How can I undelete one of my posts? What does deletion mean for a post? How do votes to delete work? What are the criteria for deletion? What else should I know about deleted posts? If I flag my question with a request to delete it, what will happen?       For more information, see the articles about  and  in the . So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive.  The laptop has windows 7 installed with ubuntu 11.04 installed through wubi.  What I want to do is remove windows completely, and make ubuntu the only OS installed on the system.  Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?)  or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in). How to tell whether two graphs are isomorphic? Why is it dangerous to initialize weights with zeros? Is there any simple example that demonstrates it? This is the proof, which I mostly understand except for one bit:  You have $h_1 \in H_1$ and $h_2 \in H_2$.   We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in h_2^{-1}H_1h_2 = H_1$. Similarly, we have $(h_1^{-1}h_2^{-1}h_1)h_2 \in H_2$. Therefore  $$ h_1^{-1}h_2^{-1}h_1h_2 \in H_1 \cap H_2 = \{1_G\} $$  and so $h_1^{-1}h_2^{-1}h_1h_2 = \{1_G\}$. Let's first multiply everything on the left by $h_1$  $$h_1^{-1}h_2^{-1}h_1h_2 = \{1_G\}$$ $$ h_1 h_1^{-1}h_2^{-1}h_1h_2 = h_1 \{1_G\}$$ $$ h_2^{-1}h_1h_2 = h_1 \{1_G\}$$   Multiply both sides on the left by $h_2$ giving us  $$ h_1h_2 = h_2 h_1 $$  The bit I don't get is right at the beginning. Why is this correct: "We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in h_2^{-1}H_1h_2 = H_1$." How can I create a water material in Cycles? Is ArcSDE no longer needed at ArcGIS 10.1? New UI encourages askers to confirm or dispute duplicate votes , the banner shown to askers whose questions have attracted at least one duplicate close vote has a couple of new options:    If the author clicks the first button, they're shown a confirmation that clarifies the results of this action somewhat:    If they click "Ok", the question is instantly closed:    If, instead, they edit, they'll be offered this guidance:    ...and once an edit is submitted the choice will go away until another duplicate close-vote is cast.   In just the past day, 80 questions  network-wide have been closed in response to confirmation from the asker; edits by askers are up significantly as well.   We'll be monitoring the results to see if further adjustments are needed. A few open questions surrounding this change include:    Perhaps if the number of votes is below some threshold?   I'm posting this separately from  to allow room for discussion and . I have start point and end point of a polyline. How can I get a point on that polyline from end points specified by the given distance.    How can I get this by using arcpy provided that this script should work on ArcView without importing any modules (). I'm trying not to hard code values.  Related Question: I am trying to install  under Wine. The program setup continually requests proxy settings. My broker tells me this shouldn't happen, and is a problem on my end. Here is what I have tried to attempt to solve this error:   I have gone into Network settings and proxy is set to none. I have looked in Wine Regedit and proxy_enable is set to 0. I have checked the firewall settings on my modem. I have tried reinstalling wine. I have installed lib32nss-mdns I can log on to my modem in iexplore using the gateway ip but thats it I've set my DNS to 10.0.0.138 to reflect the gateway I've set up a 32 bit Wine prefix   How should I proceed in solving this? Something wrong with monthly user rep - some users show quarterly figures? Can't download any app from AppStore! I'm using iPhone 6s 16 GB. My storage isn't full but still I can't download anything. Every time I wanted to start a download it keeps telling:  app name* could not be download at this time $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$ I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?  Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.  Proof: $aj=b$ and $ck=d$ for integers $j$, $k$. Then, $ac|bd$ implies $acx=ajck$ and thus $x=jk$ for some $x$. Since the product $jk$ is an integer, $x$ is an integer and thus $acx=ajck$ and thus $ac|bd$. The runabouts in Deep Space Nine sometimes have additional equipment on their backs; this is a screenshot from S02E15 (Paradise), showing the Orinoco in the background with the unknown equipment &amp; the Rio Grande in the foreground with a clean configuration:  What is the equipment &amp; why is it only sometimes present? After installing fedora and Centos and other OS now I have them all showing in my Boot menu [F2] on start up not the menu form the OS even after deleting them and reinstaling other OS - Ubuntu -so It just shows one when the OS starts but when I go to UEFI setting I still find all the old ones they instruct you to use DEL to take them out but that does not work I am new to the telecom simulations system specially multisim, which I am having a project that I want to simulate.  My project in (Analog Communications) is consisting to have FM Transmitter (Modulator) that accepts a generated signal (Sin, Cos) and get me the modulated output as a Graph.  I've found this FM transmitter circuit (3 km range):   ![two muppets][1]  and I am building that circuit on Multisim, and I failed. Because I faced many problems, first of, the variable capacitor symbol is weird here in the multisim, and the regular capacitor doesn't have variable options. Secondly, when I Use signal generator as (Sin, Cos), the output gets weird after 5 second of the simulation.  I want to show the input signal (before modulation) and the output signal (transmitted or modulated).  I guess my output result where not the as expected, as the signal (Sin with phase 90&deg;) is giving weird result of few seconds of simulating.  Below are screenshots of the simulation results within time.          Any ideas how to edit or fix the above circuit to have this output is similar to what it should be?