Prove that: $\sum\limits_{n=1}^\infty \frac{n^2}{2^n}=6$.
Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$
Onguard U-lock jammed I'm currently trying to unlock my bike that has been sitting on my porch for a while, the key works and turns, and I can hear the lock engaging/disengaging, however I cannot get the shackle to come out. I think their might be dust/sand stuck in the mechanism however I'm not sure how to get it off now to check. Find the representing matrix with the respect to the standard basis Let $\Bbb R^2$ be a finite-dimensional vector space and $T:\mathbb{R}^2\to \mathbb{R}^2$ a linear operator. Let $[T]_B =  \begin {bmatrix}0 &amp; 0 \\-1 &amp; 1 \end{bmatrix} $ be a matrix representation of $T$ in the basis $B= (1,1),(-1,0)$.   How can I find the matrix $[T]_E$ where $E$ is the standard basis $(1,0),(0,1)$ if the transformation isn't given in the question?  Thanks,  Alan In C# how do I get the list of local computer names like what one gets viewing the Network in windows explorer I sadly haven't been able to answer this questions using Google. In beamer, I'd like to make the default font for the entire title page, as well as frame titles, to be the default font from the old  (example ). After what speed air friction starts to heat up an object? I understand that air friction cools off an object at low speeds. For example, if you blow on a spoon of hot soup, it cools off. Or if you swing a hot frying pan in the air, it cools off faster.  But at higher speeds, the situation changes in the opposite. For example, consider a meteor falling on Earth. It is so fast that it heats up to such a high temperature that it burns into ashes.  What is the critical speed in which an object starts to heat up?  For example, consider a spherical object with radius of 1 meter. Let its density be 1 g/cm3 if needed. And let the air pressure be 1 atm. Assume that the temperature of the object is 400 K and the temperature of the air is 300 K. Also assume that the specific heat constant of the material is 1 cal/(gr.K). What is the critical speed for this object after which its temperature starts to rise above 500 K? Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$ Prove by Mathematical Induction . . .  $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$  I tried solving it, but I got stuck near the end . . .  a. Basis Step:      $(1)(1!) = (1+1)!-1$      $1 = (2\cdot1)-1$      $1 = 1 \checkmark$   b. Inductive Hypothesis     $1(1!) + 2(2!) + \cdot \cdot \cdot +k(k!) = (k+1)!-1$   Prove k+1 is true.     $1(1!) + 2(2!) + \cdot \cdot \cdot +(k+1)(k+1)! = (k+2)!-1$      $\big[RHS\big]$      $(k+2)!-1 = (k+2)(k+1)k!-1$      $\big[LHS\big]$      $=\underbrace{1(1!) + 2(2!) + \cdot \cdot \cdot + k(k+1)!} + (k+1)(k+1)!$ (Explicit Last Step)      $= \underbrace{(k+1)!-1}+(k+1)(k+1)!$ (Inductive Hypothesis Substitution)      $= (k+1)!-1 + (k+1)(k+1)k!$      $= (k+1)k!-1 + (k+1)^{2}k!$   My [LHS] looks nothing like my [RHS] did I do something wrong?  EDIT:     $ = (k+1)k! + (k+1)^2k! -1 $      $ = (k+1)(k!)(1 + (k+1))-1$      $ = (k+1)(k!)(k+2)-1 = (k+2)(k+1)k!-1$ Why is the subject omitted in sentences like "Thought you'd never ask"? "Thought you'd never ask" is "I thought you'd never ask" with "I" omitted. "Hope this helps" is "I hope this helps" with "I" omitted.   In English grammar, normally every sentence should have a subject, right?   My first thought is that these two examples are so often used that they are like set phrases. But these are not really set phrases. You can alter the words after "thought" and "hope".  Another possible explanation is the tendency to drop the subject if it is the first person pronoun. It seems that in many languages, such as Spanish, Italian and Japanese, the first person subject is usually omitted. Maybe English is going the same way? (Not exactly the same, since in Italian, verb forms change according to the person, so the subject is not necessary to understand who one's referring to.)  And, apparently, such omission is more common in spoken English than in written English.  Are there more examples of such first person subject omission? How frequent is it? Migrate to *any* site? Why not use the Lagrangian, instead of the Hamiltonian, in nonrelativistic QM? Undergraduate classical mechanics introduces both Lagrangians and Hamiltonians, while undergrad quantum mechanics seems to only use the Hamiltonian. But particle physics, and more generally quantum field theory seem to only use the Lagrangian, e.g. you hear about the Klein-Gordon Lagrangian, Dirac Lagrangian, Standard Model Lagrangian and so on. Why is there a mismatch here? Why does it seem like only Hamiltonians are used in undergraduate quantum mechanics, but only Lagrangians are used in quantum field theory? I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? Is the 5-foot range of the Help action my range to the ally, the enemy, or both? Is there any module or some configuration that I can define the meta tags for the views page display in drupal 7 using  3.0? Hole-and-nail paradox in special relativity How can I move files and view the progress (e.g. with a progress bar)? When moving large directories using mv, is there a way to view the progress (%)? The cp command on gentoo had a -g switch that showed the progress. It looks like we'll be adding  support to Stack Overflow. This is necessary to prevent bots, spammers, and other malicious scripted activity. We only want human beings to post or edit things here!  We'll be using a JavaScript (jQuery) CAPTCHA as a first line of defense:    The advantage of this approach is that, for most people, the CAPTCHA won't ever be visible!  However, for people with JavaScript disabled, we still need a fallback and this is where it gets tricky.  I have written a  which we can re-use.    However, I'd prefer to go with something textual to avoid the overhead of creating all these images on the server with each request.  I've seen things like..   ASCII text captcha: \/\/(_)\/\/ math puzzles: what is 7 minus 3 times 2? trivia questions: what tastes better, a toad or a popsicle?   Maybe I'm just tilting at windmills here, but I'd like to have a less resource intensive, non-image based &lt;noscript&gt; compatible CAPTCHA if possible.  Ideas? Slow math typing I have been using LaTeX for quite some time, but I find the math writing to be a bit frustrating. It seems to take me a relatively long time to code-write long math expressions, and sometimes it's hard to see where { start and end.  I am interested in writing as fast as possible.  Is code-typing each symbol really the fastest way? Is there any good program (in the spirit of Word's MathType) that enables producing math expressions by button clicking, that you recommend? How do I resize my current ubuntu partition? I've got Ubuntu 10.10 installed to use my whole hard-disk, but would like to use the 130GB of free space I've got to create a new partition and do some development on Unity and Natty.  Can I use gparted to do this?  If so how? Does the Darkness spell block vision? In discussing , I came to realize that the real question was whether Darkness blocks vision or merely creates "darkness".    Per the wording of the spell:     Magical darkness spreads from a point you choose within range to fill a 15-foot-radius sphere for the duration. The darkness spreads around corners. A creature with darkvision can’t see through this darkness, and nonmagical light can’t illuminate it. (emphasis mine)   Normal darkness is defined in the game as creating a heavily obscured area and the only description added to the darkness is that it is magical, which (to me) just means that it is created by magic and is subject to the magical rules.    Darkvision is defined in the game as basically being able to see in darkness as if they were seeing in dim light. So, the text "darkvision can’t see through this darkness" merely means that it affects darkvision in the same way it affects normal vision. (See earlier versions of the spell below, which had similar wordings.)    I see nothing there that implies it is a barrier to vision, just an active  and utter absence of illumination.    Previous editions have varying descriptions.  and  give accurate accountings of the various versions.   The 1e Wizard version seems to actually imply a sphere of opaque blackness that even blocks infra/ultravision, while the 1e Cleric version is a reversal of the Light spell and creates totally normal darkness, with no block to special visions.   The 2e version is similar to the 1e Cleric spell. It creates an area of darkness "equal to an unlit interior room".   The 3.0e version comes closest to the 5e version. It "causes an object   to radiate darkness out to a 20-foot radius. Not even creatures who can normally see in the dark (such as with darkvision) can see in an area shrouded in magical darkness." Still no reference to any sort of opacity and the wording makes it clear that darkvision is being treated specifically here.   3.5e Darkness is similar, although instead of darkness, it creates "shadowy illumination". This affects darkvision as well as normal vision.   4e doesn't have a Darkness spell, but rather a Cleric Utility called Veil of Darkness, which creates "a zone that is heavily obscured and blocks line of sight." So, finally, a reference to opacity, in "blocks line of sight", although I question if opacity was the intent. I'm getting pretty close to the reputation where I can start to edit others' posts, and I'm wondering what the etiquette for when (and when not) to do so. I figure that I should edit in the following circumstances:  Grammatical or spelling errors. Clarification where the meaning is not changed. For instance if the title does not describe the question very well. Where the user has made a very minor mistake that doesn't justify a full post to clarify. To add related resources or links that will help someone answer the post, or provide context  Before the commenting feature was added, I noticed that some moderators edit posts in order to insert a reply or extension to the post. I feel this isn't good etiquette (but I could be wrong) because I don't know what to do with their changes when I come to re-edit or respond. I also feel that I should be very careful not to change the meaning of any question or answer, even if that meaning is misguided. What other rules would be best practice?  Signatures and fluff are discouraged and are fair game to be removed.  See, for example, this post by  and this post by  and also see .  Also see Jeff Atwood's response in the comments to Getting No module name exifread error from QGIS Photo2Shape plugin?