I know that the following maths and factorials properties are useful in my problem but I don't really know how to actually get them into it a rigorous way:  $2\times 2^n=2^{n+1}$  $((n+1)!)/(n!)=n+1 $  So I did  $2((2n)!)≥2\times 2^n(n!)^2$  $2((2n)!)≥2^(n+1)(n!)^2$  But then I don't really know what to do for the $2((2n)!)$ and the $(n!)^2$; how to get them to have n+1
I am trying to use mathematical induction to prove  $$(2n)!\ge2^n(n!)^2\quad\text{for }n\in\mathbb{N}$$  I am stuck at the $n=k+1$ point.
What do bookcases do for enchanting? Are there any   editors for ? Perhaps something similar to    but for PGF/TikZ?   If there is no WYSIWYG editor for this tandem, is there one perhaps that allows the user to easily generate PGF/TikZ templates?  Alternatively, how can I most easily generate PFG/TikZ code from a visual conception? Do people mostly code their PGF/TikZ graphics from scratch? Eisenstein Criterion with a twist As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a prime, $x^{p-1}+x^{p-2}+\dots+x+1$ is irreducible over $\mathbb{Q}$? I found native people prefer to use passive voice in a sentence. For example:  Tonight's moon can be seen from anywhere worldwide. We have reason to believe that progress can be made.  Be honest, it's difficult for an English learner to understand. Why not phrase it this way:  We can see the moon from anywhere worldwide.  In which cases do people prefer to use passive voice rather than active voice? How to get Steam to recognize existing game files from a different source? Let's say I have a game bought on retail/other digital download and installed on my hard drive. I buy the same game (or a minor variant of it, say the GOTY edition) on Steam. It should be possible to use the existing installation to avoid the generally big downloads. This is very useful as it saves both bandwidth and time.   Previously, as far as I understood, Steam downloaded the game files directly to steam\steamapps\common\&lt;game_name&gt;. Since this folder would be created on starting the download, pausing and closing steam, copying the game files to this folder and then validating game files would make Steam absorb as much as possible and just download the missing files.  Currently, Steam downloads the files to steam\steamapps\downloading\&lt;game_id&gt; and then moves them over to the aforementioned folder once the download finishes. So first of all, what the final folder would be is not clear. So what would be the correct way of doing the same thing, that is, make steam use as much of the existing files as possible? Should the files be copied to downloading\*\, or should I figure out the actual folder (say via googling) and copy the files to steamapps\common\*\?  More specifically, I had a retail copy of Arkham City installed and I just bought the GOTY version on Steam. Now I tried both of the above things, in either case, the validating does not seem to do anything. Assuming I have the correct folder name steamapps\common\Batman Arkham City GOTY, according to , if I copy the files there, Steam seems to just continue with its 17GB download. If I copy it to steamapps\downloading\200260 then the download keeps stopping every few seconds with "Disk Write Error" after modifying 1 or 2 files, and I have to keep hitting resume. My company has recently exceeded our data storage cap (at 102% right now.. :/ ).  A large portion of the data that pushed us over our limit was due to an import of records.  The weird thing about this is that the records only had a couple of fields populated each out of the 200 total fields on the object.  What this tells me is that SalesForce holds the space for each field (since it knows the max-length of each field) rather than populating space only when the field is not null.  Does anyone know if this is the case?  To reiterate, and perhaps present this question more clearly:  If a field has a max character count of 1,000 and only 250 of those characters are actually being used, does SalesForce count the total data storage consumed by that field as 250 or 1,000? What desktop environments can group window icons in the taskbar like Windows 7? I find it much faster to identify a window in the taskbar (window list) by its icon,    rather than by its text:    The problem with most graphical interfaces I've  is that they don't group windows, and if you have more terminals or file managers, they'll just clutter the taskbar and you won't know which is which. Windows 7 offers a relatively elegant solution to this by grouping similar window icons together.  Question: What Linux  have this functionality? If addons exist that provide it, that's alright.  For reference, a Cinnamon applet that sort of does this is . How to install World of Warcraft? First, I'd like to apologize for the awkwardly-phrased title.   Instead of trying to explain the structure I'm thinking of, I'd like to give some context and then an example or two. I've used and seen this construction used conversationally, and if my memory and intuition serve, conversationally only. I'm curious to know if there is a formal term for it.  Ex 1  "If you don't like what I'm offering, the door is that way."  A purely logical interpretation would make little sense with this example. In this context, the inference is that the speaker is indicating to whomever he is speaking that they should leave, which suggests that there might be an elision, e.g.  "If you don't like what I'm offering, ... [then you may leave. T]...he door is that way."  Ex 2  "If you're hungry, I have food."  This is a similar context. The implication is that the speaker will offer food to whom she is speaking if they are hungry. A (strictly) logical interpretation would, again, make little sense (if you are hungry, then I have food doesn't convey any intent to provide it, nor does it make sense; the speaker always has food, she just will offer it if her guest is hungry).  I recognize that this might be too specific to have a term associated with it. I'm just curious to know what it is if it does exist. How do I technobabble without sounding like technobabble? On occasion, I've had reason to describe obviously fictional technical processes in-character. I'd like to be able to describe on these occasions without sounding like I'm saying utter gibberish. But it's not straightforward to do, particularly if I have to make stuff up on the spot.   How do I do it? Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$ Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof? I just came over this syntax in some of the questions in this forum, but Google and any other searchengine tends to block out anything but letters and number in the search so it is impossible to search out "=>".  So can anyone tell me what it is and how it is used? How does a capacitor function in a 120V AC motor circuit? I don't understand the direction of current flow when a capacitor is wired in series with the start windings in, say, a fridge or other AC motor  running at/under 120V AC. Wiring diagrams seem to suggest that voltage energizes the hot leg of the circuit and current flows through the run windings and then returns via the neutral leg. Without a capacitor, the same thing would happen to the start windings. Problem there I guess is that both windings would be in phase and thus no spin created for the rotor. So, a capacitor is wired in series with the start windings which I kind of understand. But I don't know how the voltage and current behave. Does voltage energize the hot leg, flow through the start windings and then charge the capacitor? When voltage on the hot leg oscillates down to zero, does the capacitor discharge back toward the starter windings, or does current flow through the capacitor to the neutral leg? Basically I don't understand where the capacitor's charge goes when the hot leg is hot and when it is not. Lastly, is the capacitor placed between (downstream from) the start windings and neutral, or could it be wired between incoming voltage and the windings? My players have been scheming with rapid entry tactics for the megadungeon I've recently started running. Unfortunately, the monsters are aware of the adventurers, and this has lead to a weird question of priority. Given the following situation (which I'm pretty sure is going to happen next session in some form or another) what happens?     Alice the adventurer readies an action; if her teammate Carlos opens   the door, she's going to throw the alchemists fire through the door,   which should do enough damage to kill any goblins hiding there.      Meanwhile, on the other side of the door, Bob the goblin is lying in   wait. He readies an action; if the door opens, he's going to hit the   switch to open the trapdoor in the next room, which should drop any    pesky adventurers into a pit with spikes at the bottom.      Carlos opens the door.   I know what happens when , but what happens when multiple readied actions get triggered at the same time? My guess would be an opposed reflex save or initiative check of some kind to see who goes first, but that's entirely just intuition speaking. The rulebook doesn't seem to speak to the issue. I was granted a visa to Australia few months ago. I’d like to travel to Australia in few days. However, I’ve just renewed my passport and I don’t have enough time to inform them about my passport renewal. Can I travel to Australia by carrying both my old and my new passports? I'm using IEEEtran to format an article, and I wanted to include an image. I included the graphicx and hyperref packages, and used the following code, like normal, to include an image.  \begin{figure}[!t] \centering \includegraphics[width=3in]{fetch} \caption{Pipeline: Fetch Instruction} \label{fig:fetch} \end{figure}   What appears, however, is this:    Fetch.png is a file in the directory, and if I remove it, pdflatex errors instead of producing an output. I'm using MiKTeX 2.9 on Windows 7. Any ideas? Zero visibility while editing a message in a chat room python ImportError: No module named primes For $x$ and $\lambda$ complex, if $|x| \gt \frac{1}{2} + \sqrt{\frac{5}{4} + |\lambda|}$ then $|x^2+\lambda| \gt |x| +1$