It's fairly easy to see, using the ratio test, that the series  $$\sum_{n=0}^\infty\frac{n!!}{n!}$$  converges, where $n!!$ is the double factorial.  However, I'd be interested if there is a way of calculating its value analytically. I've tried around a little with WolframAlpha and I know that   $$\sum_{n=0}^\infty\frac{n!!}{n!}\approx 4.05941$$  Wolfram also suggests a couple of possible closed forms, which look very different. Since this sum is related to $e$ somehow, the following two are what I'd find most likely:  $$\sum_{n=0}^\infty\frac{n!!}{n!}=-8-e+2e^2$$ $$\sum_{n=0}^\infty\frac{n!!}{n!}=\Gamma(e+1)+e-2-\frac{5}{2e}$$  However, I have no clue if either of the two is actually correct, and if it were, how one would go about proving this result to be true - none of the approaches to series that I know seem to be getting me anywhere.     Is there a way to find a closed form for this series, and if so, how would one go about deriving it?
My question is pretty simple.  Since $n! \gt n!!$, it's clear by the comparison test that $\sum_{n=0}^\infty \frac {1}{n!!}$ converges.      But what value does the sum converge to? How does one go about determining its value (if possible)?
Passing a formatted NSDate parskip package, keep the indents That may be a stupid question but is there a way to keep the indent while using the parskip package? I use parskip to keep spacing between my paragraphs while leaving the table of contents and lists intacts but I would like to keep the indents in my text.  Code 1 :  \usepackage{parskip}  \tableofcontents  \section{Section 1} \subsection{Sub 1}  Paragraph 1  Paragraph 2   Code 2  \setlength{\parskip}{10pt}  \tableofcontents  \section{Section 1} \subsection{Sub 1}  Paragraph 1  Paragraph 2   Basically I want the aspect of the table of contents (and possible lists) from the code 1 while keeping the indent like in the code 2. The package parskip works great but I just don't like that it deletes the indents and I'd like to find a way to prevent it from doing that.   Thank you. I have tried using glm::rotate and such. glm::translate works fine, but rotate doesn't work (if I rotate before/after a translation).  Here is my current setup:  glm::vec4 Off = glm::vec4(0, 0, 0, 1); glm::vec4 result = Util::createTransform(offset) * Off;  for (int i = 0; i &lt; Tvertex.size(); i++) {     Tvertex[i] += glm::vec3(result.x, result.y, result.z);     Tcolors[i] = currentColor; }   And here is my Util::createTransform:  glm::mat4 Util::createTransform(glm::vec3 pos) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::translate(trans, pos);     return trans; }  glm::mat4 Util::createTransform(glm::vec3 pos, glm::vec3 rot) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::rotate(trans, glm::radians(rot.x), glm::vec3(1, 0, 0));     trans = glm::rotate(trans, glm::radians(rot.y), glm::vec3(0, 1, 0));     trans = glm::rotate(trans, glm::radians(rot.z), glm::vec3(0, 0, 1));     trans = glm::translate(trans, pos);     return trans; } Rounded corner shape with rounded sides Find the domain of composition of two functions Let $$f(x)= \frac{x}{x-9} \text{ and } g(x) = \frac{-6}{x+5} $$   Find the domain of $f \circ g.$    Please show the steps and substitution/algebra to the problem. Thank you. No Sound (Dummy Output) error Ubuntu 19.10 Is the vector cross product only defined for 3D? Why is this sentence correct? “She suggested that he go to the cinema.” Why is this sentence correct?     She suggested that he go to the cinema.   I would definitely use goes instead of go. For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. Textbook recommendations for self-studying high school math? I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks that have detailed explanations and progressive practice problems, for self-study for each area such as:  Algebra  Geometry  Trigonometry and Analytic Algebra  Pre-calculus  BC Calculus  Other people have skipped grades of math due to help from tutors and parents. Can I cover all of geometry and trigonometry in 8 months without going insane and be able to skip a grade? Flash plugin up to date but Firefox keeps telling me that I have the old version Add layer to a QGIS group using Python I am trying to add new layers into a specific group which I can get to work if I know the index, but how can I get Python to find the index of a group so it can load the new layer into the group wherever it is in the TOC?  qgis.utils.iface.legendInterface().moveLayer( layer, index ) I need to find a reg ex that only allows alphanumeric. So far, everyone I try only works if the string is alphanumeric, meaning contains both a letter and a number. I just want one what would allow either and not require both. I'm sure there is a really simple explanation to this but haven't found any pointers on this forum or elsewhere. I seem to have lost the 'processing' menu from QGIS and can't figure out how/why!? Any ideas what I might have done to make this happen?  I'm running QGIS 2.8 built against GDAL 1.10 on Ubuntu 14.04. Show forms in a modal window I have a form and I am using Drupal 7. I need to show this form in a popup when a user clicks a link. The user should then be able to complete the form whilst in the pop up window. I can use anything like  or modals or anything. I just wanted to know which is the better option and what options do I have for Drupal. OAuth redirect URL is not honored Can "it" be used with plural subject? Several years ago I heard of something called dummy subjects in high school. It was then stated that, for example, it is a dummy subject when it starts many instances of sentences, e.g.        It is rainy today.   It is hard to imagine how hard is to live alone.   It is not my university degree that matters here.     My question is if we can use it with plural subjects? For example in       If some think that in Mahmoud Dowlatabadi’s works it is only social themes that matter and the writer is ignoring fictional arrays and artistic techniques, at least, endings of his stories show wrongness of this idea.     My question deals with the italicized part. If this usage is OK, can we also write they are social themes that matter? I mean can they also play the role of a dummy subject? It is generally believed and claimed that quantum computers can outperform classical devices in at least some tasks.  One of the most commonly cited examples of a problem in which quantum computers would outperform classical devices is $\text{Factoring}$, but then again, it is also not known whether $\text{Factoring}$ is also efficiently solvable with a classical computer (that is, whether $\text{Factoring}\in \text{P}$).  For other commonly cited problems in which quantum computers are known to provide an advantage, such as database search, the speedup is only polynomial.  Are there known instances of problems in which it can be shown (either proved or proved under strong computational complexity assumptions) that quantum computers would provide an exponential advantage? I have seen the word 'only' placed both before and after a noun. Where should the word 'only' go in this sentence? Is it the writer's choice?   If you have attended every lesson, only the songs need to be rehearsed at home. If you have attended every lesson, the songs only need to be rehearsed at home. Why do rainbows have distinct colors? When I searched on the Internet for the reason of formation of , I got many explanations like  one &amp; . All the explanations consider only one spherical water droplet (like ).  Why don't the refracted waves coming from the adjacent droplets interfere with each other? For example, why doesn't a red ray coincide with a violet ray coming from another droplet at a suitable position?   How are we able to see rainbows with distinct colors? Since there are thousands of raindrops in the sky when it is raining, shouldn't it be a mess due to overlapping of different wavelengths?  Please explain.