How prove this series $\sum_{n=1}^{\infty}\frac{e^n\cdot n!}{n^n}$ divergent
Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?
Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$ This winter I am planning to track my downhill skiing/snowboarding using a GPS.  Most of my riding will occur at the same resort.  I would like to be able to create a sort of "heatmap" that shows the amount of runs that I have made in a given area.  As I add more and more GPS traces to my database, my goal would be to see a sort of linear heatmap of the most traveled areas.  Given the nature of downhill skiing, you would expect the uphill chairlift lines will be the "hottest" because it will be the only places visited over-and-over again.  Given that 1) my track will not be the same every time and 2) the area covered by following one "run" may be a few hundred feet wide, what might be the best way to analyze this "linear" data to create a sort of heatmap?  My thoughts were to buffer the lines, then intersect the polys to get a sort of Venn Diagram thing going.  My preference is to use open-source technologies.  I've got QGIS and PostGIS loaded and available.  UPDATE: In regards to @blah238's response, I was thinking of something that might be able to "collect" the number of passes ("runs") through an area, and then symbolize by the count.  Conceptually, this would be similar to ArcGIS "" (but for lines, not points) or  (but for multiple line in roughly the same area).  A more visual example of a similar concept might be a traffic-flow map, where highly-congested areas would equate to "highly-traveled" ski runs/areas:    I've read the following questions that may give some ideas, but they don't really address what I am trying to accomplish: Where did the world go? I was playing on my main world when my laptop ran out of power.  I found a charger and woke it up, but Minecraft had crashed.  Now, it doesn't appear in the world selection screen!  I have verified that it still exists in the Minecraft folder, and the directory contents look to be in order.  Does Minecraft save an internal flag somewhere when it crashes hard?  If so, how do I reset it? Adding execute permission in opt folder I am trying to install oracle database in Ubuntu 16.04 LTS. I had unzipped the oracle database file in opt folder,but now when I am trying to give execute right to the sqldeveloper file using chmod +x sqldeveloper.sh, it's throwing this error:  chmod: changing permissions of 'sqldeveloper.sh': Operation not permitted Partition of a set and Hall's theorem Table with actions per row Taylor Expansion $\log(\frac{1+z}{1-z})$ I am wondering what we can say, if anything, about the population mean, $\mu$ when all I have is one measurement, $y_1$ (sample size of 1).  Obviously, we'd love to have more measurements, but we can't get them.  It seems to me that since the sample mean, $\bar{y}$, is trivially equal to $y_1$, then $E[\bar{y}]=E[y_1]=\mu$.  However, with a sample size of 1, the sample variance is undefined, and thus our confidence in using $\bar{y}$ as an estimator of $\mu$ is also undefined, correct?  Would there be any way to constrain our estimate of $\mu$ at all? Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$?    The hint says to let $H$ be the union of all $\sigma$-algebras generated by countable subsets of $\mathcal{C}$....although I don't know why. How do you create a dropdownlist from an enum in ASP.NET MVC? I'm trying to use the Html.DropDownList extension method but can't figure out how to use it with an enumeration.  Let's say I have an enumeration like this:  public enum ItemTypes {     Movie = 1,     Game = 2,     Book = 3 }   How do I go about creating a dropdown with these values using the Html.DropDownList extension method?  Or is my best bet to simply create a for loop and create the Html elements manually? C# HttpWebRequest with XML Structured Data Constructing a HoTT proof term of 1≠0 As an exercise in HoTT basics, I am trying to construct a term that has the type $Id_{Nat}(S(O),O)\to\bot$. If this were a Coq proof, I'd be done after a single inversion on the premise, as the impossible identity would leave zero cases to consider. I guess I could do something similar here, but I'm not sure how. What would be an HoTT term that has the type? There are many visa applicants who are working as an unregistered freelancer (photographer, construction contractor, fashion designer, journalist etc ). Some of these freelancers make really handsome monthly income and don't really need to work as a full time employee. Most of them receive variable amount of payments (cash, cheques, demand drafts etc) When it comes to filling a visa application form (USA, UK, Schengen etc), there is always an employment section in all visa form, eg for Schengen visa employment form :    This is a very important factor to make an application really strong.  Questions:   1 Unregistered freelancers can just simply write their profession? what documents to produce along with the application?  2 Bank statements with variable methods of payments could convince the visa officer that these are all legitimate payments? What's wrong with this equal probability solution for Monty Hall Problem? I'm confused about why we should change door in the Monty Hall Problem, when thinking from a different perspective  gives me equal probability.   Think about this first: if we have two doors, and one car behind one of them, then we have a 50/50 chance of choosing the right door.   Back to Monty Hall: after we pick a door, one door is opened and shows a goat, and the other door remains closed. Let's call the door we picked A and the  other closed door B. Now since 1 door has already been opened, our knowledge has changed such that the car can only be behind A or B. Therefore, the problem is equivalent to: given two closed doors (A and B) and one car, which door should be chosen (we know it's a 50/50 thing)? Then, not switching door = choosing A, and switching door = choosing B. Therefore, it seems that switching should be equally likely, instead of more likely.   Another way to think: no matter which door we choose from the three, we know BEFOREHAND that we can definitely open a door with a goat in the remaining two. Therefore, showing an open door with a goat reveals nothing new about which door has the car.   What's wrong with this thinking process? (Note that I know the argument why switching gives advantage, and I know experiments have been done to prove that. My question is why the above thinking, which seems legit, is actually wrong.) Thanks. Proof using Cauchy Integral Theorem Combine or rebase an arbitrarily large number of commits Issue with regex backreference in Python The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ Product norm on infinite product space LaTeX conditional expression I would like to be able to set a flag at the beginning of the main flag (say output) that tells if I want to produce a paper document or an electronic document.   Then I should check this flag to define or not empty pages, hypperref, oneside or twosides, etc.  How could I do that ?