Convergence of nth root of sum of nth potence of positive real numbers Let $k \in N$ and $a_1, a_2, ..., a_k &gt; 0$. Prove that the following series $(b_n)$ converges and define its limit.  $$b_n := \sqrt[n]{\sum_{i = 1}^ka_i^n}$$  I'm quite sure it converges to something between $min(a_i)$ and $max(a_i)$, but don't really know where to start. Sandwich theorem wasn't of much help as well.  Thanks in advance!
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?
How is the passage of time handled in Pokemon? I stopped following the Pokemon anime a while ago, but I know the series has had Ash go through many regions, including retreading at least one of them.  When Ash was about to get his eight badge, Misty mentioned that they'd been on the road for a year at that point.  I saw some recent episodes, and Ash looks nearly identical to when he first started as a trainer.  Normally I'd attribute that to cartoon sitcom logic, where the characters never age; but in a series with continuity and references to earlier seasons, this makes no sense.  What's the official word on a timeline for the Pokemon anime? In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates? [Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.]  Below, $R$ is a commutative ring with $1$.  In John J. Watkins' Topics in Commutative Ring Theory, the author observes that if $a$ and $b$ are associates in $R$, then they generate the same ideal: $(a) = (b)$. This is clearly true, because $a = ub$ and $b = va$ for some units $u$ and $v$, and so $a \in (b)$ and $b \in (a)$, whence $(a) \subset (b)$ and $(b) \subset (a)$. Indeed, we didn't even require $u$ and $v$ to be units to reach this conclusion.  The author then claims that if $R$ is not a domain, then the converse may not be true: we may have $(a) = (b)$ even if $a$ and $b$ are not associates. As a "counterexample" he gives $R = \mathbb{Z}/(6)$, with $a = 2 + (6)$ and $b = 4 + (6)$. Clearly these elements generate the same ideal: $(a) = (b) = \{0, 2 + (6), 4 + (6)\}$. The author asserts that $a$ and $b$ are not associates, but this is false: $5 + (6)$ is a unit, and $(2 + (6))(5 + (6)) = 4 + (6)$, so $a$ and $b$ actually are associates.  I figured that the author botched the example but that surely the basic fact must be true: if $R = \mathbb{Z}/(n)$ then there may be nonassociates $a,b \in R$ with $(a) = (b)$. However, I was not able to find a counterexample and eventually resorted to writing a computer program and found that there are no counterexamples in $\mathbb{Z}/(n)$ for $n \leq 100$.  So now I'm starting to think that there is no counterexample in $\mathbb{Z}/(n)$ at all, but I have not yet been able to prove this.  My questions:   If I am correct that $(a) = (b)$ in $\mathbb{Z}/(n)$ implies that $a$ and $b$ are associates, I would appreciate a nudge in the right direction toward a proof. (But if the proof is elementary, please give only a hint, not a solution :-) I'll post my solution once I have one. In either case, could you please point me to a genuine counterexample? If there is one in $\mathbb{Z}/(n)$ for some $n$, that would be ideal (no pun intended), otherwise a counterexample in any commutative ring with $1$ would be great. How to highlight source code in HTML? I want to highlight C/C++/Java/C# etc source codes in my website.  How can I do this?  Is it a CPU intensive job to highlight the source code? Find the all possible real solutions of $x^y=y^x$ Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers.  My attempt:I observed the following solutions  $x=2,y=4$  $x=4,y=2$  $y=x$  Is there any other possible solutions? Is there a good way to determine the value of magic items in 5e? In 5e, magic items aren't as common or as varied as in older editions, but that also means it is hard to place prices on them.  As DM, NPC's can't really sell a magic item because we don't know the prices for these items. While it is stated that they are rare and usually unpurchasable and unsellable, but I still have to have some basis on what to price them.  So is there a way to find buying and selling prices for magic items in 5e? Which one true in First law of thermodynamics:    $Q = \Delta U \pm W = \Delta U \pm p\Delta V$? (where $\Delta U$ is change of internal energy, $W$ work made by system and $Q=cm\Delta T $ heat made by system.)    or $\Delta U= \Delta Q + \Delta W $? (where $\Delta U$ is change of internal energy, $W$ work made by system and $Q=cm\Delta T $ heat made by system.)   First one from one the finnish physics texbook another is from . In Harry Potter, we know ghosts became ghosts when they are too afraid of death to go on - I thought about it and I'm quite certain they all go to their own version of Harry's King's Cross and become ghosts when they return as Harry did.  So can they return there? Once they've been a ghost long enough that they no longer fear death, can they move on? Counting traversed arcs on routes to write attributes using ArcGIS for Desktop? I have number of persons located on a road network and I have calculated their shortest distance to a particular location.  From the routes output created from Closest Facility can I   break the routes down into the individual arcs traversed by each person? assign each arc a count attribute indicating the number of individuals who use that arc to get to their destination?   I am using Network Analyst in ArcGIS 10.2 for Desktop. Proofs of AM-GM inequality The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way. e.g. can you link the theorem to some famous theorem, can you find a non-trivial geometric proof (I can find some of those), proofs that use theory that doesn't link to this inequality at first sight (e.g. differential equations …)?  Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics (yeah, I know, it's rather some physical argument that this theorem might be true, not really a proof), convexity, … are some of the proofs I know. “All you have to do is read” vs. “All you have to do is to read” I was speaking to an English learner and said, “All you have to do is read a lot.” And they thought that sentence wasn’t grammatically correct because I dropped the word to between is and read.  They thought it should be “All you have to do is to read a lot.” That sounds weird to me.   How do I explain to them the reason you drop the to? Or am I incorrect and it really should have a to? One might expect the sum of the tracert latencies to equal 267ms: My advisor wants me to include a name of someone who has no contribution to the paper My first advisor wants me to include my second advisor's name in a paper that I worked on before she (2nd advisor) came along.   Should I include her name even though she didn't do anything in that paper? I don't think she even read it. The entire  site is blank right now.  The header and footer are shown, but no questions. Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ I met an inequality, I ask, do not mathematical induction to prove that:     Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction Why are Flash applications so sluggish/crashy? I've noticed that Flash applications tend to be more sluggish under Ubuntu than they do under Windows on the same machine. This is particularly noticeable when watching HD video or playing graphics/physics-heavy games. Are there any ways of improving the performance of Flash under Ubuntu, or is this just an issue with the Linux version that I will have to live with?  Currently I'm just cutting down on the number of tabs open, blocking flash ads, and closing other programs, but I'm looking for ways to affect Flash itself.  Other things I have already been doing include using Youtube's HTML5 feature and playing videos straight from /tmp in VLC. I was wondering if there was some way of streamlining Flash itself though. Perhaps not.  More Specific Question: Is there anything I can do in mms.cfg to boost performance? Do virtual particles actually physically exist? I have heard  pop in and out of existence all the time, most notable being the pairs that pop out beside black holes and while one gets pulled away. But wouldn't this actually violate the conservation of energy? Picture rotation Is there a way to quickly rotate a picture in the SE editor without saving it on disk and rotating it in some image tool and reuploading it?  E.g. consider the picture below, assume I would like to quickly rotate it 90 degrees and wanted to do it expressly in the editor without the drudgery of saving it on disk, is there a shortcut in the stack exchange &quot;markup language&quot;?  If this feature is not available, I would like to nominate it for implementation. What will the universe look like for anyone falling into a black hole? I've heard that, from the perspective of an external observer, something falling into a black hole will eventually look "frozen": light waves will move to the infrared and further into lower frequencies.  But what about that unfortunate individual falling into such a black hole? What will he/she see when looking back at our universe?   For instance, I wonder, will it be the ultimate end of the Universe? Can someone who just crossed events horizon of the black hole still receive messages from our side? In Cycles, is there a way to create a procedural texture that follows the longitudinal lines of a sphere? Like a beach ball or globe? The texture should be able to be mapped to any sphere-shaped object, not just perfect spheres.    My current approach is to use a rotated wave texture, but that doesn't come from the poles (It uses latitudes) Can I get a certificate from a root CA that I can then use to sign my own web server certificates? I would, if possible, use a signed certificate as an intermediate to sign other certs.  I know that I would have to configure my systems in a certain way with "my" intermediate certificate in order to supply information about the chain of trust to my clients.   Is this possible? Are root CAs willing to sign a certificate like this? Is it expensive?  BACKGROUND  I'm familiar with the basics of SSL as it pertains to securing web traffic over HTTP. I also have a basic understanding of the way the chain of trust works, in that web traffic is secured "by default" if you encrypt with a certificate that has a valid chain all the way back to a root CA, as determined by the browser/OS vendor.  I am also aware that many of the root CAs have begun signing certificates for end users (like me) with intermediate certificates. That may require a bit more setup on my end, but otherwise, those certificates will work fine. I guess this has to do with protecting their all-valuable private key for the CA and the disaster that it would be if i were ever compromised.  EXAMPLES        Now, we are definitely not the size of any of those organizations, but they seem to be doing something like this. It would definitely make the management of these certificates a lot more palatable, especially considering one way we are expanding the reach of our e-commerce platform.