Evaluate  $\displaystyle\sum_{n=1}^{50}n \!\cdot\! n!$  I tried to write the sum like: $1+2\!\cdot\!2!+3\!\cdot\!3!+4\!\cdot\!4!+5\!\cdot\!5!+\ldots=1+4+18+96+600\ldots$  I can find a recursion like $t_{n+1}=t_n(n+2)+n!$ where $t_n=n\!\cdot\!n!$, so:  $1\!\cdot\!3+1=4$  $4\!\cdot\!4+(1\!\cdot\!2)=18$  $18\!\cdot\!5+(1\!\cdot\!2\!\cdot\!3)=96$  $96\!\cdot\!6+(1\!\cdot\!2\!\cdot\!3\!\cdot\!4)=600$  But I'm not sure how to proceed further or if this is the way to do it. Any help is appreciated.
So we learned telescoping series in class and I came across this question in my textbook and I tried to evaluate it, but I don't understand how to do it.   $$ \sum\limits_{k=1}^n\left(k \cdot k!\right) $$  According to the answers:  $$ \sum\limits_{k=1}^n\left(k \cdot k!\right) = (n+1)! - 1 $$  We learned how to simplify certain telescoping series using partial fractions but I don't have any idea on how to start this question.
How to copy all files of a certain type into a folder from command prompt? I want to copy all files of a certain type (like .docx, .pdf, etc) on my PC into a folder. How can I do that? How long does it take for police 'presence' to clear when they are alerted? I engaged with some enemy NPCs, and accidentally shot a civilian in the head (though not sure it would've mattered if I hit them somewhere else or not) as they ran across my screen. As a result of this, I was alerted by the game that the NCPD (the police) were automatically en route to 'investigate', and that I could avoid them by leaving the area and &quot;lying low for a while&quot;. How long does it take for the police to stop searching for me, and is there any kind of visual information on the HUD that alerts me to when NCPD is or is no longer looking for me? The speed of light is the same in all inertial frames.  Does it change from a non-inertial frame to another? Can it be zero?  If it is not constant in non-inertial frames, is it still bounded from above? How to guess a curve distribution from count data I have a sample composed by 2500 count data values. I've plotted in R the corresponding histogram and ecdf. I've run the One-Sample Kolmogorov-Smirnov test to check if the distribution is either exponential or poisson, but I got negative results.  Is it possible that the One-Sample Kolmogorov-Smirnov test, it's not the correct test to work with this count data.  What is the best way to identify a distribution for this kind of data? "Can/may/will you help me with this?" Which word to use when we ask for help?  Some conditions:     We know that the person asked is able to do it. We don't know if the person asked is able to do it. \phantom causes a spacing issue between paragraghs Difference between declaring variables before or in loop? I have always wondered if, in general, declaring a throw-away variable before a loop, as opposed to repeatedly inside the loop, makes any (performance) difference?  A (quite pointless) example in Java:  a) declaration before loop:  double intermediateResult; for(int i=0; i &lt; 1000; i++){     intermediateResult = i;     System.out.println(intermediateResult); }   b) declaration (repeatedly) inside loop:  for(int i=0; i &lt; 1000; i++){     double intermediateResult = i;     System.out.println(intermediateResult); }   Which one is better, a or b?   I suspect that repeated variable declaration (example b) creates more overhead in theory, but that compilers are smart enough so that it doesn't matter. Example b has the advantage of being more compact and limiting the scope of the variable to where it is used. Still, I tend to code according example a.  Edit: I am especially interested in the Java case. Max Planck, arguably the leading theoretical Physicist of his day, apparently used of experimental data to formulate his equation for , which was a major step in the birth of quantum mechanics. For this seminal event of the 20th century, I would expect a detailed explanation of how Planck arrived at his conclusion that energy MUST be quantized. I've even read his Nobel lecture but was non the wiser.  Can some kind person please explain it, or give me even just a hint? Reading Excel files from C# Does Class need to implement IEnumerable to use Foreach This is in C#, I have a class that I am using from some else's DLL. It does not implement IEnumerable but has 2 methods that pass back a IEnumerator. Is there a way I can use a foreach loop on these. The class I am using is sealed. Can md5 be used for encrypting data? I know that md5 shouldn't be used for password hashing because of collisions and possibility of making dictionary attacks e.g. using rainbow tables. But what about other uses? Can I use md5 for encrypting data?  Say, attacker gains access to database where sensitive data was stored using md5. He wants to know just this data that was hashed, not another data that gives this hash. Can he do it?  I know that there are better cryptographic hash functions so don't suggest to use them if md5 is enough. Parametric vs non-parametric machine learning methods I looked-up many references and websites and researched on how to determine if a method is between parametric or non-parametric. I came up with below definitions,     A parametric algorithm has a fixed number of parameters. In contrast, a non-parametric algorithm uses a flexible number of parameters, and the number of parameters often grows as it learns from more data.   From .  Moreover, I found,     A parametric model, we have a finite number of parameters, and in nonparametric models, the number of parameters is (potentially) infinite.    From .  And many other methods of determination, though the problem is none of them can helo ones to determine if a certain hypothetical method is parametric or not. (For instance, why k-means' number of parameters are constant but KNN is variable or basically what do we call a parameter and what we do not?) If I click to open achievements from a question when I click the X to close the achievements view I end up on the answers tab of the question, I assume because the X is over the tab and it's tapping through.    It appears if I hit the X it's ok, but if I tap anywhere else in the top row (where the X is) then I can switch to the question or answer tab. Are software titles italicized? The titles of freestanding works (books, movies, plays, albums, etc.) are italicized; smaller parts within these works (chapters in a book, articles in a magazine, songs within an album) are not (they are put in quotes, instead).  How are software titles treated?  My understanding is that if a particular program resembles a book or a dramatic work (like a video game), its title is italicized -- otherwise, it is treated as a proper noun, just like any other product.     I played Grand Theft Auto, instead of reading "Democracy in Ancient Greece" in Microsoft Encarta and preparing my presentation in PowerPoint.   Is this approach correct? I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context. Why don't the zoom buttons work on my DSLR? I am a newbie to the new camera I just bought. It is a Nikon D5200 it has the original lens that came on the camera a 18-55 lens on it. My question is when I turn the lens to the 55 and then push the plus and minus buttons at the bottom of my camera to zoom in it appears to be zooming in and then I take the photo and the picture is not zoomed in at all.    I have a new Canon T3i and was able to take and capture zoomed in photos with the lens that came with the camera (EF-S 18-55mm) as well as a Canon zoom lens EF 75-300mm 1:4-5.6. Now when I zoom in with either lens and take a photo, when I review it, it shows original image before zooming. Is it a defect in the camera or inexperience of the operator?    (Two separate questions from separate users but with the same root cause.) Why Process Builder doesn't work on Delete Scenario? As most of the time we have to think of requirement to accomplished using out-of-Box feature like e.g process builder which we think of an alternative to trigger, but suppose I have to handle scenario of Delete using Process Builder then there is limitation!(e.g calculation of roll-up on Delete) so I am curious about what is that limitation because of which Delete scenario is not handled in Process Builder. Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$. Then for all $B\in\cal B$ and $\varepsilon&gt;0$, we can find $A\in\cal A$ such that $$\mu(A\Delta B)=\mu(A\cup B)-\mu(A\cap B)&lt;\varepsilon.$$  I don't think there is a proof in this site. It's a useful result for several reasons:  We know what the algebra generated by a collection of sets is, but not what the generated $\sigma$-algebra is. The map $\rho\colon \cal B\times\cal B\to \Bbb R_+$, $\rho(A,A')=\mu(A\Delta A')$ gives a pseudo-metric on $\cal B$. This makes a link between generating for an algebra and dense for the pseudo-metric. We say that a $\sigma$-algebra is separable if it's generated by a countable class of sets. In this case, the algebra generated by this class is countable. An with the mentioned result, we can show that $L^p(\mu)$ is separable for $1\leq p&lt;\infty$, which makes a link between the two notions. In ergodic theory, we have to test mixing conditions only an a generating algebra, not on all the $\sigma$-algebra. "What have you tried" epidemic Statistics  () In Q1 of 2012 - 0.0134 such comments per question (~1.33%) In Q2 of 2012 - 0.0167 such comments per question (~1.66%) In Q3 of 2012 - 0.0169 such comments per question (~1.68%) In Q4 of 2012 - 0.0193 such comments per question (~1.92%) In Q1 of 2013 - 0.0223 such comments per question (~2.22%)  The vast majority of these questions need serious help.  Unfortunately, there is a growing trend towards abuse Along with the good, comes some bad -- as usual in life. These comments are bad when  Posting &quot;what have you tried&quot; multiple times per question is severely obnoxious Posting &quot;what have you tried&quot; when the user has clearly shown that they have tried to solve it on their own  It is becoming increasingly common to see &quot;what have you tried?&quot; comments in these scenarios. Some examples     Workarounds Flagging Comments - It is considered a good thing to be liberal with comment flagging -- that helps us clean up after the fact. &quot;Too Chatty&quot; is a no-brainer here. Unfortunately, that only cleans up after the damage has been done. All signs point toward the continued increase of the trend of using &quot;what have you tried&quot; comments. I spend a lot of time editing and cleaning up bad questions. This trend came up on my radar toward the end of last year and it is getting worse. So, I thought at the very least that I'd raise some awareness. Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure Show that the image of the curve $\Gamma\in\mathscr{C}^{1}\left([a,b]\to\mathbb{R}^{d}\right)$ has d-dimensional Lebesgue measure zero (of course, $d\geq2$).  This can be proved using the absolute continuity of $\Gamma'$ (since $[a,b]$ is compact and $\Gamma'$ is assumed continuous, hence in $L^{1}([a,b])$) together with the fundamental theorem of calculus to obtain an $\epsilon$-small cover of $\Gamma$ by balls.  But I am trying to prove this using more elementary means (i.e. without integration).  Intuitively, since $\Gamma$ is smooth, we ought to (for fine enough partitions) be able to cover $\Gamma$ by boxes which arise from its tangent line.  And by taking the partition of $[a,b]$ to be finer and finer, the "tangent box" cover ought to also get smaller and smaller.  More rigorously, the vector-version of the mean value theorem can be applied: $$|\Gamma(t_{i-1})-\Gamma(t_{i})|\leq(t_{i}-t_{i-1})|\Gamma'(t_{i}^{\star})|\leq M_{i}\Delta t$$ where $t_{i}^{\star}\in(t_{i-1},t_{i})$ and $M_{i}=\sup_{t\in[t_{i-1},t_{i}]}|\Gamma'(t)|$ which exists and is finite since $\Gamma'$ is continuous.  But to me, it's not quite clear how to rigorously construct a cover by boxes from here.  NOTE:  In the proof I mentioned (using integration), essentially you sum the left hand side over all partition intervals of uniform length $\delta$ (which depends on the $\Gamma'$), and "integrate" the right hand side.  Actually, to be more specific, for each $\epsilon&gt;0$ there exists a $\delta&gt;0$ such that $||P||&lt;\delta$ implies $\int_{t_{i-1}}^{t_{i}}|\Gamma'(t)|dt&lt;\epsilon$ (e.g. absolute continuity).  This allows you to define numbers $\epsilon_{i}=\sup_{t,\bar{t}\in[t_{i-1},t_{i}]}|\Gamma(t)-\Gamma(\bar{t})|\leq\epsilon$ so that $\sum_{i=1}^{\#P}\epsilon_{i}\leq||\Gamma'||_{L^{1}([a,b])}$.  Then you can use these $\epsilon_{i}$ to put balls at each point $\Gamma(t_{i})$ of radius (say) $2\epsilon_{i}$, thus giving you an $\epsilon$-small cover.  Again though, this is harder to establish without integration theory.