How to find the convergence of this sequence? $x_n=1+\frac{1}{x_{n-1}}$
limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$
Problem  Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. Show that $H$ is a proper normal subgroup of $G$.  So basically the result of this problem is that every group of order $2n$, with $n$ odd, is not simple.   The morphism $\phi$ can be seen as the action of $G$ on itself by left multiplication. Since $A_{2n} \lhd S_{2n}$, then $H=\phi^{-1}(A_{2n}) \lhd G$. I am stuck at showing that $H \neq \{1\},G$. All I could think of is that $G$ has an element $g$ of order $2$, then $\phi(g)$ is of order $1$ or $2$, but it is easy to see that $\phi(g)=1$ if and only if $g=1$, so $ord(\phi(g))=2$. I don't know which are the elements of order two in $S_{2n}$. I would appreciate some help to solve the problem, thanks in advance. Word for person who looks for flattery Keeping chunks loaded in minecraft I have built an iron farm somewhere far away from my actual spawn chunks in my single player world. In hindsight, this was a dumb idea, but I can't exactly change it anymore. I'd like to keep the thing running all the time, so I'm looking for a way to keep the chunks loaded.  I'm running minecraft 1.12.1, is it possible at the moment? I have looked at some chunk loader mods, but chicken chunk loader hasn't been updated yet, and the others simply didn't work.  I have tried moving the spawn chunks, but that hasn't done anything either.  Edit: I have worked this out with IronAnvil, and it ended up being a render setting in optifine which didn't load them correctly somehow. It has been working fine after I reset my optifine settings to default. How should I phrase an important question that I need to ask a professor? When needing to ask a professor about something important and potentially delicate, what is an appropriate way to phrase the question or write the email?  Examples include:   Requesting a recommendation letter Asking for exceptions to policy (e.g., ignoring a course prerequisite, rescheduling an exam) Asking if you can do research with them, join a group, get into a program Scheduling a meeting, exam, or defense Letting them know about a change of plans Letting them know about a life event affecting your work or studies (e.g., wedding, funeral, sickness, vacation) How to avoid noisy renders in Cycles? I'm trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multiple is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple $q$ of $n,m$ is a multiple of $k$, so that there exists a $g$ so that $q=k·g$.  Logically it seems correct, but I can't seem to prove it... Any help? Can I get a rebate after using my HSA? Proper subspaces of $ R^n $ I'm trying to prove that proper subspaces of $ R^n $ are closed and have empty interiors.  To prove that they are closed I'm trying to use the fact that inverse images of closed sets are closed sets by continuous functions. So I'm trying to find a linear map such that a proper subspace of $ R^n $ is its Kernel but I don't know how to express it so that it would work on all dimensions.  Regarding the fact that they have empty interiors, I really don't know what to do.  Thanks in advance. Restricting T to string and int? I have built myself a generic collection class which is defined like this.   public class StatisticItemHits&lt;T&gt;{...}   This class can be used with int and string values only. However this  public class StatisticItemHits&lt;T&gt; where T : string, int {...}   won't compile. What am I doing wrong? Is convex surface normal in cast iron skillet? I am new to cast iron skillet. I have cooked with it several times and found the Lodge 12" cast iron skillet has a convex surface which is not observable unless you pour oil on it.  When I cook, the oil move to rim of the skillet and makes the center short of oil. Is it normal?  EDIT: I added the photo of the stove and skillet for reference Prove or give a counterexample:  $$(A ∪ B = A ∪ C) \land (A ∩ B = A ∩ C) \implies B = C.$$  I think this is true, but I am not sure how to show it. I don't know if there are any manipulations with unions and intersections that allows me to move things around. Thanks in advance for any hints. Proposition II.7.5 of Hungerford's Algebra goes     Proposition 7.5. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.   Let $G = (\mathbb{Z}_6, +)$. $G$ is abelian, so it is nilpotent and thus by the proposition is the direct product of its Sylow subgroups.  The Sylow subgroups of $G$ are $H = (\mathbb{Z}_3, +)$ and $K = (\mathbb{Z}_2, +)$, so the proposition says $G = H \times K$. In particular, $0 \in H$ and $0 \in K$, so $(0,0) \in H \times K = G$... but that's not true, since $G = \{0,1,2,3,4,5\}$.  Where is/are my misunderstanding(s)? Can anyone explain precisely what IOWait is? According to HTML specs, the select tag in HTML doesn't have a readonly attribute, only a disabled attribute. So if you want to keep the user from changing the dropdown, you have to use disabled.  The only problem is that disabled HTML form inputs don't get included in the POST / GET data.  What's the best way to emulate the readonly attribute for a select tag, and still get the POST data? I'm wondering if it's possible to hide a folder without renaming it with a dot on the Desktop.  Ubuntu 12.04  Thanks Let me filter jobs by the employer's Joel Test Scores I haven't found a similar question on here, though I suspect the question may be rather well-covered.   I want to find a conformal map from the vertical strip $\{z:-1&lt;Re(z)&lt;1\}$ onto the unit disc.  Under the exponential map the region is taken to the annulus with radii $e$ and $e^{-1}$, but I'm not sure how useful this will be.  Can anyone advise on what the conformal map may be?  Thanks. How to serialize a JObject without the formatting? Density of Solid States of Compounds One of the wonderful properties of water (as my high school biology teacher would say) is that in its solid form, it is lighter than its liquid form. This means that when temperatures drop below 0 degrees Celsius, the top layer of water on, say, a lake freezes first. This works out pretty well for any fish or other aquatic creatures living underneath, because the lower layers of water will not freeze.  I know that this is an exception to the general rule of solid and liquid states: A given substance, when transformed into its solid state, will generally sink in a container of its liquid state. My question is this: What other substances are exceptions to this rule (if any?). What features do they share with water that are responsible for this? Say that Player X is 2nd in the initiative. 1st is the Black Court vampire that is in a rage and taking a swing at Player X. Player X gets a built in defense [Athletics/Fists + 4dF] but can Player X instead use his action this turn to cast his Rote Evocation Shield?  It is not his turn in the initiative order. A Block is an Action that a Player can take on his turn.   Thematically though, it seems to make sense that the char could cast a Rote Evocation Shield in response to an attack ahead of him in the initiative order. At least thematically in the books. Also, the example in YS252, shows him reacting to an attack by Agatha Hagglethorn by bring up his shield.  I am hoping there is some as yet undiscovered dark corner where this concept is covered or squashed.