find $\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$  I had $(1-x)^{-\frac{p}{q}}$ in mind. $$S=\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$$ $$S+1=1+\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$$ $$S+1=1+\frac{3}{2!\cdot3}+\frac{3\cdot(3+2)}{3!\cdot9}+\frac{3\cdot(3+2)\cdot(3+4)}{4!\cdot27}+\cdots$$ $$S+1=1+\frac{3}{2!}\left(\frac{\frac{2}{3}}{2}\right)+\frac{3\cdot(3+2)}{3!}\left(\frac{\frac{2}{3}}{2}\right)^2+\frac{3\cdot(3+2)\cdot(3+4)}{4!}\left(\frac{\frac{2}{3}}{2}\right)^3+\cdots$$ $$S+1=\left(1-\frac{2}{3}\right)^\frac{-3}{2}$$ I got $S=3\sqrt{3}-1$  But answer given is $S=3\sqrt{3}-4$
How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $$   I can see that 3 cancels out after 1/3, but what next? I can't go further.
I'm trying to prove that the characteristic of any field $F$ is either $0$ or a prime number, but I have no idea what to do.  Help? Is this a grammatically correct line in a poem: “Will he roll the dice, and follow it to Vegas?”? I want to use the following line in a poem: "Will he roll the dice, and follow it to Vegas?"   A couple of things to note; firstly, obviously I'm using "roll the dice" in both a figurative/idiomatic sense (as in taking a risk), and in a somewhat more literal sense (painting the image of the person actually following the dice that have been rolled to Las Vegas). That being said, my confusion here is if I can get away with using "follow it" or if I need to say "follow them" .... As I've read online "dice" traditionally refers to more than one die (i.e., the plural of die), but in the modern usage, "dice" can also be used to refer to the singular (i.e., just one die).   So, it seems like if I was referring to the image of just one die, then the way I phrased it could be correct. However, the image I'm trying to paint is the one usually associated with the phrase "roll the dice" (i.e., rolling two dice)...   So could "follow it" still be considered acceptable in this sense... or would I need to use "follow them"?... And if not referring to the dice themselves, could using, could the "it" be taken to refer to the act of rolling the dice, or would that not work either? Thanks so much in advance!! I have a simular issue to the question asked in :  I have tried the solution provided by Jake. This makes that my picture fails to resize (scale) as i intend.  Whether i give Set scale=0.4 or Set scale=0.8 The size of the  picture remains the same  This is the code i use:  \begin{figure}[ht!] \centering \begin{tikzpicture}[Set scale=0.4, every node/.append style={transform shape},] % STYLES \tikzstyle{every node}=[rectangle,rounded corners, minimum width=100pt,     align=center,node distance=3cm,fill=black!10,inner sep=5pt,text width=3cm,minimum height=1.75cm,&gt;=stealth,text badly centered] % Draw forces \node [force, above of=rivalry] (top) {Firm strategy, structure and rivalry}; \node [force, left=1cm of rivalry] (left) {Factor conditions}; \node [force, right=1cm of rivalry] (right) {Demand Conditions}; \node [force, below of=rivalry] (bottom) {Related and Supporting industries}; %%%%%%%%%%%%%%%% % Draw the links between forces \path[&lt;-&gt;,thick]  (left) edge (right) (left) edge (top) (left) edge (bottom) (top) edge (right) (bottom) edge (right); \path[&lt;-&gt;,thick]  (bottom) edge (top); \end{tikzpicture} \caption{Porter's Diamond source:~\cite{Porter:1980}}% \label{fig:diamond}  \end{figure} How do ideal quotients behave with respect to localization? Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When does localization at $S$ commute with taking quotients, i.e. when does the equality $$(S^{-1} J : S^{-1} I) = S^{-1} (J:I)$$ hold? More generally, when is it true that $$S^{-1} \text{Ann}_{R} (M) = \text{Ann}_{S^{-1} R} (S^{-1} M)$$ for an $R$-module $M$? What should I do if a user posts sensitive information as part of a question or answer? I've observed several incidents where users inadvertently posted detailed, sensitive information. For example, a Server Fault user who posted a question with working login credentials to his actual production server, or the Stack Overflow user who posted actual medical records as example data for his question.   What should be the procedure for handling such cases? Do ISTQB/ISEB Testing Certificates prove that someone can test? I have passed the , and to be honest, I'm pretty sure that anyone could pass that exam with a couple of days of studying. My question is are these certificates really of value to my career and should I put any effort into acquiring the ? Access Java / Servlet / JSP / JSTL / EL variables in JavaScript Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. One hour Lufthansa transit time in Frankfurt from Bangalore to Dublin How to install Hp laser jet 1020 plus printer? Class imbalance strategies When dealing with the class imbalance problem in a binary classifier, there are three ways I know of to address it: over-sampling, under-sampling and using cost-sensitive methods.   Are there any guidelines, rules of thumb or general strategies to choose among these methods? A possible answer would be: over-sample when the positive class has more than 100 instances (I just made it up). Solidify thickness doesn't correspond to specified scale I'm Having trouble with setting the thickness applied by solidify. The selected block is 15 by 28 cm (single mesh sides) and the thickness of solidify is set to 3mm (pointing inwards). Checking off even thickness makes it even worse. I was using the function in a different object earlier today and it worked just fine. Is there something I'm missing or is this just a bug?   Feel free to treat me like an idiot, I've literally started using blender yesterday :) Show address bar w. full path in Nautilus instead of breadcrumbs bar How do I audit my reputation? If I am concerned that my reputation score is incorrect, how can I audit it, or get a report of a detailed breakdown of my reputation? How do I set focus follows mouse? Is there a way to set up "focus follows mouse" behavior in Unity? UK coastline shapefile? How can I calculate magnification ratio I get by reversing a lens? I am looking for a way to mathematically determine the maximum magnification ratio that is possible/practical to achieve by reverse mounting a non-macro lens, prior to purchasing the lens.     See related questions: Why is this trigger finding a null value despite the field being populated? Financial Account is a child of Client in a Master-Detail relationship. The client (John Sandbox) has a Wealth Advisor, 'John Smith', on his client record, however, the system debug that pulls finAcct.client__r.Wealth_Advisor__c shows this field as 'null'.  Does anyone have any idea why I am not getting the value in this field? The goal of this trigger is to make sure that the lookup fields for the Client Team (one of which is 'Wealth Advisor') always match between the client record and the financial account record when they are linked. Thanks!      Boolean UpdateMe;     List &lt;Financial_Account__c&gt; FinAcctsToUpdate = new List&lt;Financial_Account__c&gt;();      //Loop through all Financial Accounts in Trigger.new     for(Financial_Account__c finAcct : newFinAccts){         UpdateMe = false;          System.debug('finAcct WA Lookup: ' + finAcct.Wealth_Advisor_Lookup__c);         System.debug('Client: ' + finAcct.client__c);         System.debug('Acct WA: ' + finAcct.client__r.Wealth_Advisor__c);          if (finAcct.Wealth_Advisor_Lookup__c != finAcct.client__r.Wealth_Advisor__c){             finAcct.Wealth_Advisor_Lookup__c = finAcct.client__r.Wealth_Advisor__c;             UpdateMe = true;         }         if(UpdateMe == true){             FinAcctsToUpdate.add(finAcct);         }         UpdateMe = false;     }      if(FinAcctsToUpdate.size() &gt; 0 ){         update FinAcctsToUpdate;     } Adjective for someone who exaggerates emotions to get sympathy What's the word for someone who tries to get sympathy from others by exaggerating their emotions or constantly reminding them of something tragic that happened to them?  I'm looking for an adjective - no disorders or health related definitions please. Automatically hide old comments Comments are useful. Many, many posts need some small amount of meta discussion, and comments fulfill this need.   Comments are also noisy and distracting. Occasionally, someone will post an insightful and informative comment that complements the post its attached to... But more often, they're tangential, quickly irrelevant and in the way: stale comments show up first, making newer comments less visible.  The guidance here for a long time has been that comments are transient, . And yet, some do hang around for months or years because .   So let's fix that.   I propose that all comments older than 7 days be hidden by default unless the comment has been up-voted in the past 30 days. Hidden comments should be indicated by a note below the post ("add / show 3 more comments") and made visible to anyone who cares to click it.   This doesn't destroy anything, but it would get it out of the way of most readers - and hopefully encourage folks to edit it into the actual answers as a result.  Note: I previously suggested this . Other  have been . However, I'm not suggesting that comments be automatically deleted, and they should be hidden based purely on the age / activity of the comment itself, not of the post to which they're attached.