Question:   Let $\{a_n\}$ be a sequence defined by $$a_1 = 2, a_{n+1} = \frac{1}{3 - a_n}, \text{ if } n \geq 1$$  Does $\{a_n\}$ converge or diverge?  My solution:  I just wrote down some terms.  $a_1 = 2, a_2 = 1, a_3 = \frac{1}{2}, a_4 = 5/13, a_5 = 13/34$. Basically eventually $a_n$ will be zero, so I say this converges to $\frac{1}{3}$. What is the most simple but a good solution to prove this?
Let $\{a_n\}$ be defined as follows: $$a_1 = 2, a_{n+1}=\dfrac{1}{3-a_n}$$ $n \geq 1$.   Does $\{a_n\}$  converge?    Using the monotone convergence thereom, if the sequence is both bounded below and decreasing, $\forall n\in\mathbb{N}, n\geq 1$, then we can say the sequence converges.  I'm having trouble proving that the sequence is bounded below by induction.    Claim: The sequence is bounded below by $0$.  Base Case:  $n=1$  $a_1 = 2 \geq 0$, so this holds.  Induction Step  Suppose that $a_k \geq 0$, for some $k \in\mathbb{N}$. (IH)  Prove $a_k \geq 0\to a_{k+1} \geq 0$  I have some difficulty here.  $a_{k}=\dfrac{1}{3-a_{k-1}} \geq 0$  $a_{k+1}=\dfrac{1}{3-a_{k}}$, but we know $a_k \geq 0$, but we don't know how much bigger. It could be that $a_k = 4$, and then I have $a_{k+1}=\dfrac{1}{3-4}$, which is negative, and does not confirm with my claim of being bounded below by $0$. Have I gone somewhere wrong in my induction?
Function that returns the second argument of Reap I have a fraction with sums above and under the line. How can I convince LaTeX to write the indices of the sums under the sigma instead of next to it?  \begin{displaymath}   \frac{\sum_{s \in S} s^2}{\sum_{p \in P} p^2} \end{displaymath} How do I unset an element in an array in javascript? Long boot delay on Ubuntu loading/splash screen following regular dist-upgrade on clean SSD install (18.04) Minecraft looks like a bad texture pack I am playing Minecraft 1.12 but today the in-game textures look much lower quality than usual. I don't have any of the super secret settings turned on, and I'm unsure how to fix this and get the normal textures back. how to place JS-code to NewForm.aspx of a list in SharePoint 2013 APP.  In sharepoint 2010 we can use this list definition to add JS:  &lt;List ...&gt;   &lt;MetaData&gt;     ...     &lt;Form Type="NewForm" Url="NewForm.aspx" SetupPath="pages\form.aspx" WebPartZoneID="Main" UseDefaultListFormWebPart="False"&gt;         &lt;WebParts&gt;           &lt;AllUsersWebPart WebPartZoneID="Main" WebPartOrder="1"&gt;             &lt;![CDATA[              &lt;WebPart xmlns="http://schemas.microsoft.com/WebPart/v2"                 xmlns:sfwp="http://schemas.microsoft.com/WebPart/v2/SimpleForm"&gt;               &lt;Assembly&gt;Microsoft.SharePoint, Version=15.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c&lt;/Assembly&gt;                                        &lt;TypeName&gt;Microsoft.SharePoint.WebPartPages.SimpleFormWebPart&lt;/TypeName&gt;               &lt;FrameType&gt;None&lt;/FrameType&gt;               &lt;Title&gt;JS1&lt;/Title&gt;               &lt;sfwp:Content&gt;ANY HTML YOU NEED HERE               &lt;/sfwp:Content&gt;             &lt;/WebPart&gt;             ]]&gt;           &lt;/AllUsersWebPart&gt;           &lt;AllUsersWebPart WebPartZoneID="Main" WebPartOrder="2"&gt;             &lt;![CDATA[             &lt;WebPart xmlns="http://schemas.microsoft.com/WebPart/v2"&gt;               &lt;Assembly&gt;Microsoft.SharePoint, Version=15.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c&lt;/Assembly&gt;                                        &lt;TypeName&gt;Microsoft.SharePoint.WebPartPages.ListFormWebPart&lt;/TypeName&gt;               &lt;PageType&gt;PAGE_NEWFORM&lt;/PageType&gt;             &lt;/WebPart&gt;             ]]&gt;           &lt;/AllUsersWebPart&gt;         &lt;/WebParts&gt;       &lt;/Form&gt;     &lt;/Forms&gt;   &lt;/MetaData&gt; &lt;/List&gt;   How to do the same for Sharepoint 2013? I am trying to send an email using a Visualforce template and I want that template to include an icon. Is this possible? What is the true origin of the orcs? There are a bunch of theories about the true origin of the orcs. Tolkien tried out a few different origins for his Orcs throughout his life but died before he could fully revise The Silmarillion with his final view on their origins and nature.   There is a list of origins proposed by Tolkien():   Made of stone and slime through the sorcery of Morgoth; East Elves, where The Silmarillion contains a suggestion that Orcs are descended from East Elves captured by Melkor, their minds and bodies distorted and corrupted. Sentient beasts, another of Tolkien's theories proposes that Orcs may have begun as soulless animals of vaguely humanoid shapes, empowered by the will of the Dark Lord (Morgoth) and learning language only as parrots do. Fallen Maiar, there are hints in the History of Middle-earth series of books, (especially in Morgoth's Ring in the section "Myths Transformed"), that some Orc leaders, such as the First Age's Boldog, or the Great Goblin encountered by Bilbo and the Dwarves, may in fact have been fallen Maiar which had taken Orc form. Corrupted Men, while Tolkien at some point saw all Orcs as descended from the original corrupted and tortured Elves, later comments of his indicate, according to Christopher Tolkien in Morgoth's Ring "Myths Transformed" text X, that he began to feel uncomfortable with this theory. At about the same time he removed the references to the Thrall-Ñoldorin, he also began searching for a new origin for the Orcs. It seems Tolkien wanted to change the origin of the Orcs to make them corrupted and twisted Humans.  A mix of corrupted Elves and Men, A late idea of Tolkien seems to be that Orcs (or Orks, showing the late spelling change) had a mixed origin of Elves and Men. Some cross-bred with Men, Tolkien also suggested that Men were cross-bred with Orcs under Morgoth's lieutenant, Sauron (and possibly under Morgoth himself). The process was later repeated during the War of the Ring by Saruman. This possibly refers to the way the Uruk-hai and the Half-Orcs were created, in The Lord of the Rings.   Of all these theories, which is the most accepted, and which did Tolkien agree with the most? I always thought that orcs were made of East Elves, as explained in The Lord of The Rings, but the story goes beyond this, there are the stories in many other books that explain different things.  What can you tell me about this? How to convert QString to std::string? What is the difference between a HMAC and a hash of data? Are split infinitives grammatically incorrect, or are they valid constructs? Mark's generosity in this crisis seems to more than make up for his earlier stinginess.   Should those sentences always be avoided, or are there cases where they are valid? Merge two unadjacent partitions I have three partitions on my drive, arranged something like the following:   ___________ | 1 |  2 | 3| -------------   I wish to merge partitions 1 and 3.  Is there a way to do this, or do I have to manually "slide" partition 2 over to partition 3, and then merge the newly open space where partition 2 used to be?  Note: I do not have the space to copy partition 2 to partition 3 (2 is 200GB and 3 is 65GB). If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Inverse Laplace transform of $\frac{1}{s\sinh{c\sqrt{s}}}$ I have some questions.  I have to take the inverse Laplace transform of the given function below by residue theorem:   $\frac{1}{s\sinh{c\sqrt{s}}}$   where s is Laplacian variable and c is a real constant. Actually it has a very crowded numerator which I do not need to define. So far I know that it has a simple pole at s = zero and infinitely many poles at c√s = nπi  Questions:  1- What is the order of the poles at nπi? Is it 1/2?  2- At zero, what is the order? (Because to residue at zero, s also contributes)  3- What is the way of solving this question at nπi by residue theorem (without defining sinh function as exponential function)?  Thanks for any help. A real solution to ambiguous tags Introduction Even within a site's relatively focused range of topics (it's unlikely anybody on SO will think  refers to the solar system's star), some tags are inherently ambiguous. There are currently two unsatisfying options:  Declare that the tag is only to be used for one of the possible meanings, write a tag wiki explaining it, and link from there to the newly created tags for the other meanings, and keep cleaning up the tag whenever someone ignores the tag wiki popup explaining the tag. Use alternative tags for all meanings (e.g. cd-command and compact-disk, instead of cd), and keep removing the undesired tag whenever it pops up again. And it will. The  is not deterring users from creating unnecessary or redundant tags.  Blacklisting tags is a complicated ritual occurring once in a blue moon, at most, and involves a dragon, a team member, and sacrifices to be burnt. It also makes those other, unambiguous tags not easier to discover, if it were applied to ambiguous tags. I don't consider this to be a viable option in this case. Proposal Here's the proposed : Just like a tag can be declared a synonym of another tag, basically changing all references to the synonym to references to that other, canonical tag, it should be possible to make it a disambiguation tag (or ambiguous tag) for multiple other tags. It's a logical progression from &quot;You entered a term with the same meaning as this other term&quot; to &quot;You entered a term with the same meaning as one of these other terms, please choose the correct meaning&quot;. Use of the tag should force the user to choose one of the alternatives presented:  &quot;, did you mean  or ?&quot; &quot;, did you mean , , or ?&quot; &quot;You entered 'vm', did you mean  or ?&quot; &quot;You entered 'sas', did you mean  or ?&quot; &quot;You entered 'kvm', did you mean  or ?&quot; &quot;You entered 'mono', did you mean  or ?&quot; &quot;You entered 'dsl', did you mean  or ?&quot;  Not all of these examples might be good (I'm not sure e.g. about the first two SO examples), but that's not really the point here anyway — they should just convey how this could be used. Note how some of these tags would probably never be used if users had to type them, and yet everything shorter is ambiguous. This could probably be mostly solved by extending the synonym/tag wiki excerpt dropdowns shown when a user enters the tags, proposing the alternatives for an ambiguous tag entered by the user (like a synonym suggestion, but with multiple options). If ignored, the user could still be shown an error message like with blacklisted tags, but by designing the selection dialog well, it should be possible to prevent that in most cases.  Related topics   This topic is related, but the user seems to have accepted the inclusion of tag wiki excerpts in the tag suggestions drop-down as sufficient.   This requires extensive discussion and team involvement (i.e. the status quo)   This topic has no real solution, except manual retagging. How can I separate a separable function Ability to retract reopen/delete votes It seems that now we can retract close votes but we cannot retract reopen votes or delete votes.  I am trying to retract my reopen vote  but it does not allow me to.  Can we have that ability? A linear operator commuting with all such operators is a scalar multiple of the identity. Subscribe to a particular question by email (not RSS)