Evaluate : $$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$  Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried using  and , but to no avail. An elementary solution is appreciated. Thanks in advance.
I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it.
My computer only has one USB-C port and I want to connect multiple USB-C devices to it.  I have trawled through Amazon but am amazed that I cannot find a single one. There are many hubs that have USB 3.1 USB-A ports but not USB-C ones.  Is there a limitation in the spec that prevents this being possible? If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian? I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample. How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$ Prove ${n \choose p} \equiv \left\lfloor \frac{n}{p} \right\rfloor \pmod p$   ($n$ is natural and $p$ is a prime number)  How can i prove this statement? Find and Replace several several different values all at once After a brief discussion with @MonicaCellio (In the comment) - It has come to my attention that perhaps I chose the wrong words to describe my original idea (In the edit history). When I wrote "social features" I was actually suggesting an alternative way to for users to share their knowledge using a new type of post and articles and not just by questions and answers.   The basics of my idea was to allow users to write articles on topics relevant to the community on the SE websites just as any answer posted is expected  to be too. Allowing users to write their own content (I.E posts/articles/etc..) will be granted as a privilege for advanced users so that not anyone could write stupid things, users that will write content must be established and well-familiar with the rules of SE.   I know anyone can start a blog, but it's require frequent maintenance and content updates, while writing on SE ensure that it addressed to a large, well-established community, with a similar fields of interest.  Posts writing will follow the same guidelines as asking and answering questions does, but it will provide an alternative for users that are mostly answering specific questions, as a new method to contribute to the community.   By writing, one may also reveal more about their personality, wit, way of thinking, ETC. That's when the "social" takes place - You'll also have "authors" and not just "users" you can remember by their catchy name, but people that you can remember by the field of interest they are writing content about.   They key points will be:    Users will be able write rich content with advanced WYSIWYG HTML editor, content that doesn't require a question to answer, but that is still relevant to the SE website it is writing on. Users will be able to Upvote/Downvote other users posts. Posts will be listed on their author profile, so it will not just show the activity, but additional layer of user generated content. Popular posts with high reputation could be associated to tags making the information of tags richer with user generated content Users will be able to include links in answers not only to other questions, but also to posts. User will be able to comment on post to allow the author to correct and improve it. It will improve the interaction and offline discussions (Which are prohibited when it come to long discussions on questions and answers). I believe that users will be happy to share their own tutorials, tips and tricks - Because they are already doing so when answering questions, but before the can start, they'll have to earn enough reputation to ensure that only high profile users will be able to write high profile content, and not just any rubbish. It will provide another type of method for voting and getting earning reputation Users will be able to include content in their CVs on SO Jobs.  Opportunity to earn some new badges.   So it's a Mega feature request and it require some detailed design work, but I do think that it will invite a new type of activity for users on SO (and the rests of the sites of-course). Javascript parseInt() with leading zeros Kill some processes by .exe file name Best method for short time-series I have a question related to modeling short time-series. It is not a question , but how. What method would you recommend for modeling (very) short time-series (say of length $T \leq 20$)? By "best" I mean here the most robust one, that is the least prone to errors due the fact of limited numbers of observations. With short series single observations could influence the forecast, so the method should provide a cautious estimate of errors and possible variability connected to the forecast. I am generally interested in univariate time-series but it would be also interesting to know about other methods. Where is the cron / crontab log? Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$ Is there anyway to get tuple operations in Python to work like this:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (4,4,4)   instead of:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (1,2,3,3,2,1)   I know it works like that because the __add__ and __mul__ methods are defined to work like that. So the only way would be to redefine them? Create bootable CD from ISO Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other? I have two tensors that i must calculate double dot product. matrix A is rank 2  and matrix B is rank 4. I want to multiply them with Matlab and I know in Matlab it becomes: A : B = trace (A*B) but it has one error and it says: Inner matrix dimensions must agree So how can I solve this problem? anybody help me? Thanks How is light affected by gravity? Light is clearly affected by gravity, just think about a black hole, but light supposedly has no mass and gravity only affects objects with mass.  On the other hand, if light does have mass then doesn't mass become infinitely larger the closer to the speed of light an object travels. So this would result in light have an infinite mass which is impossible.  Any explanations? How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $? How do I show via inductive proof (and without the use of the binomial theorem) that $$ {n\choose k} \in {\mathbb{N}} $$ $$\text{ for  all }\: n,k \in \mathbb{N}_0\: \text{ with }\: n≥k .$$ For my induction proof, I would show that$$  {n\choose k-1}+{n\choose k} = {n+1 \choose k}   $$ is true. However, how do I start the induction correctly with the base cases and the hypothesis? As far as I know, I cannot simply assume $$ {n\choose k-1}+{n\choose k}, $$  so how do I come to this derivation? How can I determine if it is possible to check luggage during a layover? A common desire among some travelers with long layovers is to leave the airport for a while, and perhaps do some shopping, or even buy goods in the airport.  But this then often necessitates the ability to check additional luggage during the layover, for the remainder of the journey.  There might be other reasons to want to check luggage mid-journey as well.  Are there any general rules one can use to determine whether this is an option in a given airport when traveling with a given airline, and the associated costs (when it is possible)? Find the Values of $p$ and $q$ such that $\int_0^{\infty} x^p\ln(1+x)^q dx$ Converges or Diverges. The question is to find the value values of $p$ and $q$ such that the improper integral converges or diverges.  My friend indeed found some values of $p$ and $q$ such that it converges, but after many trials, I am still getting divergent for all values.  Via L'Hospital, we have the following:  $$\lim_{x \to 0} x^c\ln(1+x)^q = \infty \space \text{when} \space  c &lt; 0$$ $$\lim_{x \to 0} x^c\ln(1+x)^q = 0 \space \text{when} \space  c &gt; 0$$ $$\lim_{x \to \infty} x^c\ln(1+x)^q = \infty \space \text{when} \space  c &gt; 0$$ $$\lim_{x \to \infty} x^c\ln(1+x)^q = 0 \space \text{when} \space  c &lt; 0$$  Thus, $$\int_1^{\infty} x^p\ln(1+x)^q dx = \int_1^{\infty} x^{p-c}x^c\ln(1+x)^q dx &gt; \int_1^{\infty} x^{p-c} dx$$ which diverges when $p &gt; -1$.  Here, we set $c = \frac{1}{2}(p+1) &gt; 0$.  On the other hand, $$\int_0^1 x^p\ln(1+x)^q dx = \int_0^1 x^{p-c}x^c\ln(1+x)^q dx &gt; \int_0^1 x^{p-c} dx$$ which diverges when $p &lt; -1$.  Here, we set $c = \frac{1}{2}(p+1) &lt; 0$.  Thus, the integral diverges whenever $p \ne –1$.  Then, let $p = -1$.  $$\int_0^{\infty} \frac{1}{x}\ln(1+x)^q dx = \int_0^1 \frac{1}{x}\ln(1+x)^q dx + \int_1^{\infty} \frac{1}{x}\ln(1+x)^q dx $$ $$ \ge \int_0^1 \frac{1}{x+1}\ln(1+x)^q dx + \int_1^{\infty} \frac{1}{x+1}\ln(1+x)^q dx $$  The first summand diverges when $q &lt; -1$ and the second diverges when $q &gt; -1$.  Finally, let $ p = -1$ and $q = -1$.  The integral $\int_0^{\infty} \frac{1}{x\ln(1+x)} dx ≥ \int_0^{\infty} \frac{1}{(x+1)\ln(1+x)} dx$ diverges on both ends.  Did I make a mistake? Add days to JavaScript Date How to add days to current Date using JavaScript. Does JavaScript have a built in function like .Net's AddDay? Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$.