Prove by induction that $b_n-a_n < (\frac {1}2)^n(b-a)$
prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
Often called "He-Who-Must-Not-Be-Named" and "You know who", was it common knowledge at the Ministry of Magic that Voldemort was the same person as Tom Riddle? How to sanity check a date in Java I find it curious that the most obvious way to create Date objects in Java has been deprecated and appears to have been "substituted" with a not so obvious to use lenient calendar.  How do you check that a date, given as a combination of day, month, and year, is a valid date?  For instance, 2008-02-31 (as in yyyy-mm-dd) would be an invalid date. What is the equation for the scale factor of the universe, a(t), for the best fit of data to the $\Lambda CDM$ Model of Cosmology? Ideally I like a single formula or multiple formulas for different time ranges that would cover the time from the end of inflation through 100+ billion years after the big bang using  the $\Lambda CDM$ Model.  I know that from the end of inflation back to the time of the big bang would be much more speculative, but some wild estimate would be appreciated for that time range also!  The Friedmann–Lemaître–Robertson–Walker metric defines the scale factor, a(t), from the metric: $$-c^2d\tau^2 = -c^2dt^2+a(t)^2d\Sigma^2$$ where $d\Sigma^2$ ranges over the 3 dimensional space of the universe and does not depend on time. Usually the scale of the scale factor is set by defining  $a(t_{now}) = a(13.78 B yr) \equiv 1$ Why are mutable structs “evil”? Following the discussions here on SO I already read several times the remark that mutable structs are “evil” (like in the answer to this ).  What's the actual problem with mutability and structs in C#? 14.04 microphone noice buzzing sounds My Microphone started acting up badly this morning has a loud buzz, it has got to be software it was working fine last night installed updates and shutdown last night and boom here we are, oh and i have a Microphone jack microphone and a internal mic, both are acting up... Please help  PLEASE help i need to use skype.  Dafull97 How do I use multirow in LaTeX? A zero-dimensional ring is Noetherian? Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian.   Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$, contradiction. Hence every prime ideal is maximal. Consequently, every increasing (or decreasing) chain of prime ideals will be trivial in the sense that it will have length $0$. Hence the ring is Noetherian (and Artinian).  Is there a problem with the above proof? When should I use GET or POST method? What's the difference between them? Does any skill have a global cooldown? I am new to this game, and will try to optimize my game-play as much as possible. I will do this by making macros for my spells and abilities.  Are there global cooldown on spells and abilities in League of Legends?  Since this is important regarding macros to trigger spell and ability combos.  A Global Cooldown(GCD) is a cooldown which affect most if not all abilities, spells or items when using an ability, spell or item.   In most games it is there to prevent macroing/spamming of multiple instant abilities, spells or items, and usually has a set short duration independent of the ability, spell or items normal Cooldown. Suppose $a_n \ge 0$ for all natural numbers $\Bbb{N}$ and $\sum{a_n}$ converges. Is it necessarily true that $\sum{(a_n)^2}$ converges?  I think yes.  Clearly as $a_n \ge 0$ it follows $(a_1+a_2+\dots+a_n)^2 \ge a_1^2+a_2^2+\dots+a_n^2 \ge 0$ and so $(\sum_{n=1}^{\infty}{a_n})^2 \ge \sum_{n=1}^{\infty}{a_n^2} \ge 0$.  As we are told $\sum{a_n}$ converges to say $L$ where $L$ is a real number it follows $L^2 \ge \sum_{n=1}^{\infty}{a_n^2} \ge 0$ and so therefore $\sum{(a_n)^2}$ converges.  Is this correct? Working with large numbers in PHP What's the right course of action for cross-site duplicates?  There are two flavors:   A question is posted which belongs on another site and ends up already having been asked there. I can't find an example right now, but it's bound to happen soon if it hasn't already. The same question is posted in two places at the same time. Take  for example, which is a duplicate of , posted within a half hour of each other by the same user.   Possible actions:   Move it to the proper site and close it once it gets there as a duplicate. Provide a way to close it right away as a cross-site duplicate, without moving it first. Flag it and let a moderator merge them. Close it as "no longer relevant" or something else.   Additionally, flavor 2 is a bit of a special case; perhaps the question really spans both sites in terms of content and intended audience. Should multiple-postings in this case be allowed? If so, should they somehow be linked to each other? If not, how do we avoid a migration war? I have a log file being written by another process which I want to watch for changes. Each time a change occurs I'd like to read the new data in to do some processing on it.  What's the best way to do this? I was hoping there'd be some sort of hook from the PyWin32 library. I've found the win32file.FindNextChangeNotification function but have no idea how to ask it to watch a specific file.  If anyone's done anything like this I'd be really grateful to hear how...  [Edit] I should have mentioned that I was after a solution that doesn't require polling.  [Edit] Curses! It seems this doesn't work over a mapped network drive. I'm guessing windows doesn't 'hear' any updates to the file the way it does on a local disk. How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$ How to show  $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$ After swapping the positions of the hour and the minute hand, when will a clock still give a valid time? At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and the minute hand position does a clock still give a valid time? Does an Animagus's animal form change along with their Patronus? In the two examples shown of an Animagus capable of casting a corporeal Patronus (Minerva McGonagall and James Potter) both had the same animal as their Patronus and as their Animagus animal.  Now, it is also shown that extreme circumstances can change one's Patronus. I think the only example of this was Tonks, which turned from a jackrabbit into a wolf in response for her love of Lupin.  If Tonks (or any other wizard who had their Patronus change) had been an animagus, would her animagus have changed as well? Are any fictional worlds available in some GIS format? Is there any fictional world (e.g. ,  etc.) which could be used in a GIS context?  For example, an MBTiles archive and a shapefile or any other format usable with OpenLayers / Polymaps / Leaflet? Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities? Complex polynomial of degree $n$