Using Sandwich Theorem, how can I evaluate the limit for -  $$\lim_{n\to \infty} (a^n+b^n)^{1/n}$$ where $a$ and $b$ are positive quantities and $b$ is greater than $a$.
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 can I extract menthol powder from fresh mint/peppermint? How do I use the conditional (ternary) operator? I've always wondered how to write the "A ? B : C" syntax in a C++ compatible language.  I think it works something like: (Pseudo code)  If A &gt; B    C = A Else    C = B   Will any veteran C++ programmer please help me out? I am looking for an adjective/noun for someone who is able to speak/ makes speeches very well and uses words effectively to influence/impress people. Tools to get a pictorial function call graph of code I have a large work space which has many source files of C code. Although I can see the functions called from a function in MS VS2005 using the Object browser, and in MSVC 6.0 also, this only shows functions called from a particular function in a non-graphical kind of display. Additionally, it does not show the function called starting from say main(), and then the functions called from it, and so on, deeper inside to the leaf level function.  I need a tool which will give me a function call graph pictorially with functions callee and caller connected by arrows or something like that, starting from main() to the last level of function, or at least showing a call graph of all functions in one C source file pictorially. It would be great if I could print this graph.  Any good tools to do that (need not be free tools)? Custom fields not useable in Approve/Reject Approval Request I'm trying to get through my first SalesForce mod and it's definitely been a rocky path. I have one task I'm trying to do, but this section is holding me back.  I have two custom fields.  One a pick list and the other a text box. I have added them to the Approve/Reject Approval Request section in the Opportunity page. I added the fields through the Select Fields to Display on Approval Page Layout and they DO show up on the Approve/Reject Approval Request section. I added them in like ().  However, just the label for the fields show, the field itself is nowhere to be found and can't be used.    You can see and even use the fields when you open the Opportunity page itself, but when you go to use the Approve/Reject, you can't do anything with them or see any values.  I was never a SalesForce user and got this tossed in my lap and really don't know my way around yet. I appreciate any help on this!!  Thanks!!  Kurt When used like this:  import static com.showboy.Myclass;  public class Anotherclass{}   what's the difference between import static com.showboy.Myclass and import com.showboy.Myclass? A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n &lt;&lt; 1$.  Does this approximation have a name?  Any leads on estimating the error of the approximation? Session_ID_Remove Issue in API callouts in Salesforce Upvoting a downvoted answer will do +2 upvote? I have first downvoted an answer. Then, I decided to leave it the way it was before, meaning to undo my downvote just like it never existed, so I upvote to remove my downvote. Suprise! SO seems to consider it as two upvotes!  Let's say I have an answer with -3, then I decide to downvote it making it -4. After deeper thoughts, I prefer to remove my downvote to let it the way it was before I downvoted, meaning -3, but SO makes it -2 automatically!     Wouldn't it be supposed -3, and not -2? If someone else would have upvoted, it would have made it an upvote +1, not a +2!   Am I missing something here?  In fact, is there a way to tell SO that I only wish to undo my downvote, and not upvote the answer? I may understand why it does behave this way. I only wish to undo my downvote, though. How can I select a point without selecting its reflected side Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$? Reading about photons I hear different explanations like "elementary particle", "probability cloud", "energy quanta" and so forth. Since probably no one has ever seen a photon (if "seen" it supposedly - and rather conveniently - ceases to exist) but many experiments seem to verify its properties (or are they maybe adjusted to fit the experiment). I thus can't help wondering if the "photon" is then just a physical/mathematical tool with inexplicable properties (like zero mass - but affected by gravity fields - and constant speed c in space) invented to explain some otherwise unexplainable phenomem and to supplement the elementary particles and their interactions. In short: Are they real or imaginary!  Does anybody know? Or maybe the answer is "blowing in the wind" because to most physicist it probably just does'nt matter as long as it works (as the alternative healers say). Sorry if I seem a little sarcastic here and there. Is there a black hole in the centre of the Milky Way? Finding total after percentage has been used? Tried my best with the title. Ok, earlier while I was on break from work (I have low level math, and want to be more fond of mathematics)  I work retail, and 20% of taxes are taken out, and I am wanting to find out how much I made before 20% is taken out.  So I did some scribbling, and this is without googling so please be gentle. but lets say I make 2000 dollars per paycheck, so to see how much money I get after taxes I do 2000 * .8 which gives me 1600. I can also do 2000 * .2 which gives me 400 and then do 2000 - 400 (imo I don't like this way since I have a extra unnecessary step)  Anyways, I was pondering how to reverse that to have the answer be 2000, and what I did was 1600 * 1.20 (120%) which gave me 1920, and I thought that is odd so I did 1600 * 1.25 and answer was 2000 exact.  My question is why did I have to add extra 5% (25%) to get my answer? I am sure I did something wrong, and I fairly confident the formula I am using is a big no no.  edit; wow thank you all for your detailed answers. I am starting to like mathematics more and more. Prove: $$x^\alpha-\alpha x \le 1- \alpha  \\ \forall x\ge 0 \ ,  \ 0&lt;\alpha &lt;1$$   It does resamble Lagrange's MVT, in order to get to the RHS, say $\alpha\in[0,1]\Rightarrow \frac{1-\alpha-0^\alpha-\alpha 0}{1-0}=1-\alpha$.  But I'm not sure what to do about the LHS. I always get a negative value in the LHS and we know that the RHS will always be postive but that doesn't really help with proving it...  Edit: maybe define a function: $g(x) = x^\alpha-\alpha x - 1+ \alpha$ Derive it: $g'(x) = \alpha x^{\alpha-1}-\alpha $ but then I get to $x^{\alpha-1}=0$    Also, I had two questions when trying to figure this out:   Can we use deriviation of both sides of an inequality to prove it ? Can we use limit of both sides of an inequaltiy to prove it ?   Thanks. How to succeed with making omelette My application listens on port 8180.  When I am trying to connect to port 8180 locally. Connection got refused. But when I am trying to connect to port with ip . I am able to connect.  [root@MUMLGOMAPP002 ~]# telnet 10.209.129.54 8180 Trying 10.209.129.54... Connected to MUMLGOMAPP002 (10.209.129.54). Escape character is '^]'. ^] telnet&gt; q Connection closed.   [root@MUMLGOMAPP002 ~]# telnet localhost 8180 Trying 127.0.0.1... telnet: connect to address 127.0.0.1: Connection refused telnet: Unable to connect to remote host: Connection refused   My interface is :  [root@MUMLGOMAPP002 ~]# ifconfig bond0     Link encap:Ethernet  HWaddr 44:1E:A1:73:F7:50           inet addr:10.209.129.54  Bcast:10.209.129.63  Mask:255.255.255.224           UP BROADCAST RUNNING MASTER MULTICAST  MTU:1500  Metric:1           RX packets:322997568 errors:23 dropped:0 overruns:0 frame:0           TX packets:304470664 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:213628580915 (198.9 GiB)  TX bytes:217449664400 (202.5 GiB)   I would like to know what is the difference between telnetting localhost and ip. ‘And what on earth’s a Squib?’ said Harry.   To his surprise, Ron stifled a snigger.   ‘Well – it’s not funny really – but as it’s Filch ...’ he said. ‘A Squib is someone who was born into a wizarding family but hasn’t got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual. If Filch’s trying to learn magic from a Kwikspell course, I reckon he must be a Squib. It would explain a lot. Like why he hates students so much.’ Ron gave a satisfied smile. ‘He’s bitter.’   Chamber of Secrets - pages 110-111 - British Hardcover   If a Squib is 'kind of the opposite of a Muggle-born wizards' wouldn't that make a Squib more Muggle than wizard? Does a Muggle strictly have to be born to Muggle parents to be considered a Muggle, or can the lack of magic render a person, even of wizarding parentage, a Muggle?  Hermione can say in all truthfulness that she is a 'Muggle-born witch.' The nickname (albeit derogatory) for a Muggle-born is 'Mudblood.' Filch can say in all truthfulness that he is a 'wizarding-born non-magic person,' which, according to Hagrid is a Muggle.  'Squib' may be derogatory in the same way as 'Mudblood' is. Neville laments his family thought he was a Squib when he showed no magical tendencies as a child; the Longbottoms were so rattled at the thought of Neville being a Squib that they, among other things, threw him out a second-story window to see if he would bounce. So could 'Squib' actually be a derogatory term meaning the opposite of 'Mudblood'? After all, the Ministry doesn't allow Squibs to register as magical beings, and many Squibs are schooled and trained in Muggle professions. Squibs don't get a Hogwarts letter.    Wizards can be born to Muggles. Can Muggles be born to wizards?  ETA 11.22.14: In regard to this question, I was glancing through Philosopher's Stone, and I came across this, that Neville said:     ‘Well, my gran brought me up and she’s a witch,’ said Neville, ‘but the family thought I was all Muggle for ages. My great-uncle Algie kept trying to catch me off my guard and force some magic out of me – he pushed me off the end of Blackpool pier once, I nearly drowned – but nothing happened until I was eight.’   Philosopher's Stone - page 93 - Bloomsbury - chapter seven, The Sorting Hat   Could what Neville says contribute to the theory and/or conclusion that Squibs are, or are not, Muggles? How do we apply what Neville said? How to align prices in columns when both 2 digit and 4 digit variations of a price exist? I am trying to display data similar to coinmarketcap with prices for coins, some above 1$ in which case they are rounded to 2 zeroes, some below 1$ in which case they have 4 digits following the decimal. I saw a bunch of posts here where people have suggested to use right alignment for numeric applications/prices  Approach 1 Decimal aligned   1. $13,700.05  2.      $0.0048   Approach 2 Right aligned   1. $13700.05  2.   $0.0048   Which of these approaches do you suggest? Thank you for your time Is a constant condition more costly than switching shaders? In general, branching in shaders is not a good idea. But now I have a shader with a condition that is constant with respect to the entire draw call. So the branch that is executed is always the same for one draw call.  Is such kind of branching still more costly than having multiple shaders without these branches and switch between them? Gray cone obscuring/clipping the 3D Viewport