I know this is a series of an arithmetic and geometric progression product which looks like  $$\sum\limits_{n=0}^{\infty} (2n+1) \left(\frac{1}{2}\right)^n$$  but I don't know how to calculate the sum. Any help would be appreciated!
I'm trying to find the infinite sum that is defined by:  $$ 3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots $$  However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!
Probability of complete coupon set in $K$ boxes where a box has $N$ distinct coupons? How do I turn off the arrow for desktop icons? Best way to obtain clarifying information when reputation doesn't allow comments? Date manipulation Function SSJS I want to delete the records from Data Extension based on Date Field in my DE i.e. I want to delete record which were created date before 30 days.But I didn't find date manipulation function in SSJS like dateadd() in ampscript Convert web page to image I'll be at my uncle's house just in case you need to reach me.      I'll be at my uncle's house should you need to reach me.   Could you tell me what is the difference between those? Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance. What is the need of complex functions in wave analysis? It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$y=y_{_0}e^{i(kx-\omega{t})}$$ where $y_{_0}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular velocity and $x$ &amp; $t$ are position an time respectively. Ofcourse, I know that the function $e^{ix}$ can be written in the form $\cos{x}+i\sin{x}$ and so it is a periodic function with period $2\pi$ but my question is for what purpose we define it in terms of complex numbers? It seem to be more convenient to use real functions for real variables such as amplitude, electric and magnetic field of an electro magnetic wave, and also in quantum mechanics. What actually this interpretation means or what is the advantage of such functions? How do I access the list of currently logged on users through Terminal Services Manager in Windows Server 2012? Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what  "really" is?  Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics  approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses.   I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics. Where can I find 32-bit version of Ubuntu? I need to find the 32-bit version, and all I keep finding is the 64-bit. Can anyone provide me with a link? jQuery serialize does not register checkboxes Slow leak on front cruiser tire I ride my cruiser to the supermarket and back, using either a trailer or a basket over the front wheel for carrying groceries. The bike has developed a persistent slow leak on the front wheel, and I've replaced the tube without it helping. I can't see anything puncturing the tire, and the spokes aren't poking through the rim tape that I can see.   These are 26" wheels with low pressure (around 40 - 50 psi). What else can I look for that might be causing this? Prove that there exist two integers such that i - j is divisible by n. Here's the full question:  Prove that, for any $n + 1$ integers, $\{x_0, x_1, x_2, . . . , x_n\}$, there exist two integers $x_i$ and $x_j$ with $i \neq j$ such that $x_i − x_j$ is divisible by $n$.  Now, the integers aren't necessarily consecutive, positive, or without repeats. I've tried breaking this into cases, such as "$x_i = x_j, \dfrac{0}{n} = 0$" etc. But I don't think it's possible to exhaust the cases.  I feel like there should be an easy way to contradict this one, by saying that $x_i - x_j$ isn't divisible by $n$. But I just wasn't getting anywhere with that one.  I thought about using induction, but I would have just as hard a time proving this for $n$ integers before $n+1$ integers, so I don't think it would be possible for me to do it that way.  If anyone could help out, that would be wonderful. I'm pretty stuck here. Could "them" mean "those"? Background Nowadays, I see &quot;them&quot; used to mean &quot;those&quot; a lot. I don't know if it was as common in the past. For example, take &quot;one of &quot;. On researching about it, I found some people say it comes from a dialect of British English. Some others say it is a &quot;non-standard&quot; usage. I see this usage in Canadian English also, and it seems some people use it in a sarcastic way. Moreover, I have seen a song titled &quot;one of them days&quot;. I also read, in the book called &quot;A Broken Promise&quot;, &quot;&quot; Finally, Wikipedia says that it is a usage in  (a common name for the Southern Midland dialect of American English):  Pronouns and demonstratives &quot;Them&quot; is sometimes used in place of &quot;those&quot; as a demonstrative in both nominative and oblique constructions. Examples are &quot;Them are the pants I want&quot; and &quot;Give me some of them crackers.&quot;    Question(s): What would you say about the usage of this word? Is it correct? Could we use it in daily speech? Could this usage go beyond a specific dialect and be used in other dialects, regions, etc.? Does it really originate from Appalachian English? Why did this usage become popular among other English speakers? Note: I have already seen a similar question:  However, it only says, &quot;ungrammatical,&quot; there. This question is specific to this situation only, and there is more to it. Extracting one band from three band raster in ArcMap? How can I quickly get a player back into a session after their character dies? As a GM, what kind of techniques could I use to help a player re-join the gaming session after their character dies? I want to avoid any major interruptions to the flow of the adventure. I would also like to avoid having to end the session immediately or making them sit on the side-lines for the rest of the session. I am programming with normal pointers, but I have heard about libraries like Boost that implement smart pointers. I have also seen that in Ogre3D rendering engine there is a deep use of shared pointers.  What exactly is the difference between the three, and should I stick on using just a type of them? Where is the .Trash folder? What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?  Some subjects off the top of my head that probably need covering:   Differential geometry, Manifolds, etc. Lie groups, Lie algebras and their representation theory. Algebraic topology.