For me, multiplication is a binary operation so it can be applied only on a finite sequence of numbers. but $1^{\infty}$ requires that we apply multiplication infinitly which is not defined as multiplication is a binary operation.  Is that a good reason? If not, what is the reason?   If my reason is ok, So similarly, $5^{\infty}$ is indeterminate ?   ِAdded:   I noticed that all answers are in context of "limits". Algebraically, multiplication is a binary operation, So it ONLY can be used to define multiplication of finite sequence of numbers as not a infinite sequence. So algebraically, what does $1^{\infty}$ even mean?
From : In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form.    The indeterminate forms include $0^{0},\frac{0}{0},(\infty - \infty),1^{\infty}, \ \text{etc}\cdots$   My question is can anyone give me a nice explanation of why $1^{\infty}$ is considered to be an indeterminate form? Because, i don't see any justification of this fact. I am still perplexed.
How do I convert a light circuit with a single pole switch to use two 3-way switches? I have a box in the hall which includes a 3-way switch (with another switch at the other end of the hall), a single pole switch to the garage lights and a single pole switch to the exterior lights on the garage.  The single pole switch to the garage is wired with a 3 wire plus ground. The white and black wires feed the garage lights from the switch and the red wire feeds the garage ceiling outlets and is hot all the time.  In the garage I would like to add a 3-way switch for the garage lights and another 3-way switch for the exterior garage lights.  In other words I would like to be able to turn on both of those light locations from both the inside hall and the garage as well. I know that I will of course have to replace the existing single pole switches in the hall with 3-way switches and add a junction box in the garage with 2 3-way switches in it.  The thing that I don't know is what wiring connections I need to make.  Can you help? I was wondering if quantum particles do actually  exists in two different states simultaneously and if it has been proven they do indeed exists in a superposition of states.  How has it been figured out since observing it would collapse the wave-function into one single state (two superposition of states into one ) as it has been mentioned in the Schrodinger's cat experiment? Creating a DateTime in a specific Time Zone in c# I'm trying to create a unit test to test the case for when the timezone changes on a machine because it has been incorrectly set and then corrected.  In the test I need to be able to create DateTime objects in a none local time zone to ensure that people running the test can do so successfully irrespective of where they are located.  From what I can see from the DateTime constructor I can set the TimeZone to be either the local timezone, the UTC timezone or not specified.  How do I create a DateTime with a specific timezone like PST? Argument list too long error for rm, cp, mv commands By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.  Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.  For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.  So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex? We have a few cases of chat accounts remaining without valid parent account:  and  for example. What should mods do when that happens:  delete the chat account notify the SE team so they can reattach or clean it up leave it as is: it's a normal situation Is there an idiom for people who boast too much? How do gas molecules constantly bounce without losing energy? From a related question (), i asked myself this: How do gas molecules constantly bounce off each other without losing energy?  If you drop a ball, it bounces a bunch of times, but the height of each bounce gets shorter and shorter. Because it loses energy in each bounce.  So I don't see how a pressurized gas doesn't lose energy. As I understand it, pressure is a manifestation of lots of molecules bouncing off each other. So over time, shouldn't the pressure drop as the molecules lose speed? and eventually all the molecules will settle in a pile on the floor.  I think another way to phrase this is, how do elastic collisions not lose any energy in the exchange? My understanding of the 2nd law of thermodynamics is that some energy is always "lost" when it's converted from one form to another, or transferred from one object to another. I.e., no transfer/conversion of energy is ever 100% efficient. Is there a way to disable recovery mode so it won't be possible to factory reset your phone?  Basically what I am trying to do, is that it should be impossible to factory reset the phone as there is no recovery partition.  (Example format the recovery partition, or hide the option of factory reset on the recovery menu, build a custom recovery etc.)  P.S.. I was thinking of maybe removing a system app (example package installer) and making it a user app, and therefore when you factory reset the phone it won't start cause the system app is missing, but when I tried that, the phone wouldn't start even without resetting it, as the system needs the system app to start, ( I thought that maybe even though I removed the app from being a system app, the phone would still start as it's still there as a user app, and only after resetting the phone - that the user apps get deleted, the phone won't start as it's missing the app).  Thanks in advance Cycles not rendering image or movie clip used as background in the viewport How to make nautilus use file-roller in 17.10 like in 17.04? In Ubuntu 17.10 ubuntu upgraded Nautilus (File Browser) to version 3.26.0. New nautilus is not using file-roller anymore, it has  to an integrated compression mechanism.     This makes it complicated to create encrypted archives and/or different types compression archives that file-roller does provide.  How to make nautilus use file-roller as before?  Related question: How to reach local computer via machine name instead of IP? Developer here prototyping a mobile app, somewhat new to mobile:   I have a simple HTTP service running (currently) on my main laptop with machine name latitude. Another laptop named vaio sharing the same WiFi network can access that service via http://latitude:8080 -- in other words, it can easily resolve the (current) IP 192.168.x.y (or whatever dynamic IP it has at that moment) of latitude somehow. Now my HTC Android phones, connected to the same WiFi network, can reach the IP address of latitude so opening in its browser http://192.168.x.y:8080 works. But I'd like to use the machine name here, too, like when I access the HTTP service from vaio, simply by typing http://latitude:8080 and have the phone resolve the current dynamic IP of latitude. This doesn't work, only get the standard browser error page: "Webpage not available".   Is there some DNS setting in Android I need to tweak? Haven't found anything from playing around with Settings, and a question like this is notoriously hard to google, too...  The One V has Android 4.0, and the One M7 has Android 4.2 in case that makes a difference.  (Note, this should work even when the Android is not USB-connected to my computers and is not acting as a WiFi Hotspot -- obviously, since it's consuming my main WiFi network anyway). SGS2 screen gets more and more dark blue, possibly after overheated charging How can I split an equation over two (or more) lines I am having the following equation:  \begin{equation}   Q(\lambda,\hat{\lambda}) = -\frac{1}{2} P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) \left( n \log(2 \pi ) + \log \left| C_m^{(s)} \right| + \left( \mathbf{o}_t - \hat{\mu}_m^{(s)} \right) ^T C_m^{(s)-1} \left(\mathbf{o}_t - \hat{\mu}_m^{(s)}\right) \right) \end{equation}   which does not very well fit on one line. How can I split this over two lines? What I have in mind is that I specify the splitting place, and that the first line is left aligned and the second line right aligned to make clear that it is still the same equation.  The linebreak \\ does not work. Please let me know how to remove certain repositories in Ubuntu 11.04. I am getting the following errors while updating.  sudo apt-get update  Err http://ppa.launchpad.net natty/main Sources                                      404  Not Found Err http://ppa.launchpad.net natty/main i386 Packages                                404  Not Found ......  E: Some index files failed to download. They have been ignored, or old ones used instead. I can’t get used to the new Unity system. I’d like to move the left Unity panel to the bottom of screen (like Windows 7 – more similar is better).  But I can't find a way. Is it possible? Change font of formula number in DisplayFormulaNumbered Can you use external money to pay trading commissions in tax-free and tax-deferred accounts? Gravitational waves and time What is Interface method in salesforce