I have to show that $a_n=\sqrt[n]4$ is convergent and then find the limiting value. How can I formally show that an is convergent? To find the limiting value is it enough to use this line of reasoning:  $\sqrt[n]4=4^{\frac{1}{n}}$ and then $\frac{1}{n}\rightarrow0$ for $n\rightarrow\infty$ so $\sqrt[n]4=4^{\frac{1}{n}}\rightarrow1$ for $n\rightarrow\infty$ ?
I am looking for a proof possibly using the sandwich theorem and/or Bernoulli's inequality for proving the following statement:  If $x&gt;0$ then $x^{1/n}$ tends to $1$.
Why constructors will always have same name as of class and how they are invoked implicitly? Why is DarkGray lighter than Gray? Book identification about boy who gets computer zapped into head Modified files in a git branch are spilling over into another branch Nonabelian semidirect products of order $pq$? Following an earlier question ,  I would love to implement the  as suggested by Russell McMahon.   That way I could switch between 4AA batteries (1.2V*4 or 1.5V*4) and USB power. The battery voltage would nicely be regulated to 5V too. And the idea was to use the shutdown pin of the chip IF USB power is detected. The device also has to work when no batteries are present.  I am not too experienced with this, but I thought to connect the 5V from the USB to the SHUTDOWN pin from the LDO chip. Pitywise, the pin function is inverted, if SHUTDOWN is low, Vout from the LDO chip is low.  I am now considering using an XOR gate to make the thing work:         &ndash; Schematic created using   I would love to limit the amount of components, but the number of diodes is already high, and the whole thing feels a bit crafted   My questions:    Is there a better suitable chip (with an inverted shutdown pin) is there a better circuit (without voltage drop) to get this (I would prefer to get rid of D1, but then, the XOR has no desired operation) The problem is stated as follows:  "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective."  Regardless of the right solution, I don't understand why is the following wrong:     We have $\theta: R\to R$. By the isomorphism theorem $R/\ker\theta\cong\operatorname{Im}\theta$. Since $\operatorname{Im}\theta = R$, it follows $\ker\theta =0$, so it's injective.   Harsh criticism will be appreciated. Thanks. I opened Cairo-dock in my terminal with: open cairo-dock and that worked but if I close the terminal it goes with it. How can I keep it open regardless of the terminal? What are the survival limitations of alcohol? This question was inspired by watching one of the Pirates of the Caribbean movies; particularly where Jack Sparrow allegedly survives on a desert island by finding an unlimited supply of rum. I've heard alcoholic drinks dehydrate you more than hydrate you, so this seems like an unsustainable strategy.  My question is threefold.    How long can a human survive by drinking hard liquor (40% alcohol by volume) for their sole water intake (ignoring food)? If it's unsustainable, what will finally take you down? If the hard-liquor approach isn't viable, what % alcohol can keep you hydrated (beer, wine, etc.)?   Related: 1.  2. What is the difference between == and === in php. I am unsure when to use both.  Updated note: So that it shows up in StackOverflow search, the difference between == and === is the same as the difference between != and !==. Blender resizing my image in half I have a problem. I'm trying to render an image and the resolution is 500x280. After I render the image and save it, for some reason the image gets resized to 250x140. I'm unsure why it happends. When transfering data using TLS the browser and server agree the cipher suite to be used - so for example this could be chosen as AES-128 and is (probably) outside of my control. If I separately encrypt a file using AES-256 and then Alice downloads this via her browser (using AES 128) have I still effectively got AES-256 security?  My reasoning goes like this: even if the AES-128 encryption on the TLS link was broken by Eve then all she would get is the AES-256 encrypted file and she would then have to break that encryption to see the plaintext.  Of course the AES-256 key is sent to Alice over a separate channel. I have a bit of code where I am looping through all the select boxes on a page and binding a .hover event to them to do a bit of twiddling with their width on mouse on/off.  This happens on page ready and works just fine.  The problem I have is that any select boxes I add via Ajax or DOM after the initial loop won't have the event bound.  I have found this plugin (), but before I add another 5k to my pages with a plugin, I want to see if anyone knows a way to do this, either with jQuery directly or by another option. Is there a difference between Bayesian and Classical sufficiency? The title pretty much says it all. I wonder whether there is any difference in the way Bayesians understand sufficiency vs. the way orthodox statistics understands sufficiency, or are they equivalent?  If there is a difference, what is it? I took a look at  but my question is not the same.  I have unique keys encrypting each plaintext (in CBC mode, AES-256) and I do not use a key to encrypt more than one plaintext. Is it insecure to use null IVs in this case? If it is insecure, why? Let $A =$ {$(x,y,z) \in \mathbb{R}^3 |$  $x,y,z \geq 0$ and $x+y+z \leq 1$}.   Compute the integral $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$  using new variables $u = x+y+z$, $v=\frac{y+z}{x+y+z}$, $w=\frac{z}{y+z}$.  Also, although it was not stated in the problem, I'm assuming that $m,n,p,q \in \mathbb{Z}$.   Since $f(x,y,z)=(x+y+z, \frac{y+z}{x+y+z}, \frac{z}{y+z}) = (u,v,w)$, $f^{-1}(u,v,w) = (u(1-v), uv(1-w), uvw)$  Thus J $= \begin{vmatrix} 1-v &amp; -u &amp; 0\\  v(1-w) &amp; u(1-w) &amp; -uv\\  vw &amp; uw &amp; uv\\ \end{vmatrix} = u^2v$  Skipping the part where I find the new limits of integration, and skipping some algebraic simplification we have:  $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$   $$= \int_{0}^1 \int_{0}^1 \int_{0}^1 u^{m+n+p+2}(1-u)^q v^{n+p+1}(1-v)^mw^p(1-w)^n du dv dw$$  However, I have no  idea where to go from here. Any help would be much appreciated. Number of bitstrings with $000$ as substring I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$?  Now there are many  ways to go about this but if I choose starting a $n-3$ bit string and append it with $000$ how would I show that? doing comparison if else in JasperReports Do DOM tree elements with ids become global variables? Backquote in lsblk output What does ` mean in this output from lsblk?