I know there are very many threads about this question already and I have perused them to make sure this is unique.    The quantity $0^\infty$ perfectly well defined.  Therefore, when we have the expression $0\times\infty$, what error is invoked when I do the manipulation  $$ 0\times\infty= 0^\infty\times 2^\infty=(0\times2)^\infty=0^\infty=0~~?$$  The above is one case where I have conjured $2$ from "the ether," so let me give another example where I don't have to "deconstruct" infinity as I have with $\infty=2^\infty$.  Let's say I have an infinite product function of an extended real number (reals including infinity) so that the domain is $x\in[-\infty,\infty]$:  $$ f(x) = \prod_{k=1}^\infty \dfrac{(1+k)^x}{x}  $$  It follows that  $$ f(\infty) = \prod_{k=1}^\infty \dfrac{(1+k)^\infty}{\infty}=  \prod_{k=1}^\infty \bigg[0\times (1+k)^\infty  \bigg] $$  Since the least value of $k$ is $1$, every term $(1+k)^\infty$ is going to be equal to infinity and, if I evaluate it, I will get $f(\infty) = \Pi (0\times\infty)=$ undefined.  However, before I evaluate the $k$ term, I can use $0=0^\infty$, and then write   $$ f(\infty)  =  \prod_{k=1}^\infty \bigg[0^\infty\times (1+k)^\infty  \bigg] =  \prod_{k=1}^\infty \bigg[  [0\times (1+k)]^\infty  \bigg]=0 $$  Since there is a way for me to do the operations which does not lead to an undefined expression, is this order of operations preferred.  Any insight much appreciated.  EDIT: This question differs from the one it is alleged to duplicated because it asks questions about exponent operations that do not appear in the alleged duplicate question.  Furthermore, the question about the structure of the infinite product is completely disconnected from the alleged duplicate question, other than it does it relate to the same parent issue $0\times\infty$.
I did a bit of math at school and it seems like an easy one - what am I missing?  $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$  $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ 0}_{n\text{ times}} = 0$$  (i.e add $0$ to $0$ as many times as you like, result is $0$)  So I thought an infinite number of $0$'s cannot be anything but $0$? But someone claims different but couldn't offer a reasonable explanation why. Google results seemed a bit iffy on the subject - hopefully this question will change that.
Meaning of symbols $\vdash$ and $ \models$ I'm confused about the use of symbols $\vdash$ and  $ \models$. Reading the answers to  and  I see that:  the turnstile symbol $ \vdash $ denotes syntactic implication. Then $ S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$  the double turnstile $ \models $, denotes semantic implication. Then the notation $A \models B$ means simply that B is true in every model in which A is true.  So I understand that, e.g., $(ZFC+ A) \vdash B$ means that $A \Rightarrow B$ is a theorem in ZFC and, in this sense $\vdash$ is a mathematical symbol for a theorem. While $(ZFC+ A) \models B$ means that $ B$ is true in every model of ZFC in which $A$ is true, and, in this sense $\models$ is not a mathematical but a metamathematical symbol.  I understand correctly? And those are the only uses of the two symbols? When did UX start being called UX? When did people start talking about User Experience as a role/domain similar to the way that it's used today?  The earliest reference I can find is the Norman, Miller, Henderson CHI 95 piece on Apple's set up (Donald Norman, Jim Miller, Austin Henderson: What You See, Some of What's in the Future, And How We Go About Doing It: HI at Apple Computer. Proceedings of CHI 1995, Denver, Colorado, USA)  Anything earlier?  [Edit - after @Andrew's nice answer we have 1993 as an earlier date     "I invented the term because I thought Human Interface and usability   were too narrow: I wanted to cover all aspects of the person's   experience with a system, including industrial design, graphics, the   interface, the physical interaction, and the manual." - Don Norman   ]  (I'm writing a talk on communities of practice relating to UX - and want to get my time lines right ;-) How can a lens with a single focal length focus on more than one plane? $f$ has the form $f(x) =ax^2+bx+c$. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal? What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE? How to prove that $\lim\limits_{h \to 0} \frac{a^h - 1}{h} = \ln a$ In order to find the derivative of a exponential function, on its general form $a^x$ by the definition, I used limits.  $\begin{align*} \frac{d}{dx} a^x &amp; = \lim_{h \to 0} \left [ \frac{a^{x+h}-a^x}{h} \right ]\\  \\ &amp; =\lim_{h \to 0} \left [ \frac{a^x \cdot a^h-a^x}{h} \right ] \\  \\  &amp;=\lim_{h \to 0} \left [ \frac{a^x \cdot (a^h-1)}{h} \right ] \\  \\ &amp;=a^x \cdot \lim_{h \to 0} \left [\frac {a^h-1}{h} \right ] \end{align*}$  I know that this last limit is equal to $\ln(a)$ but how can I prove it by using basic Algebra and Exponential and Logarithms properties? Thanks checking apt-get update lock file How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago? I was reading through the Wikipedia article on  and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the  exactly says      The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.   Now I come to understand that the  singularity is believed to be around 13.8 Billion years ago.  So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?   I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?  Edit: My Bad! Here's how..  A simple google search led me to  which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Cursor hitting "border" between differently sized monitors I'm not sure if I'll be to explain this right. I have a dual monitor display on Windows 7 and my left monitor (laptop) has a lower vertical resolution than the right monitor (regular LCD monitor). So obviously, if I stick these monitors together, there's a part in the upper-left corner of my LCD monitor where cursor cannot go to the other monitor. I remember there was a trick to make this gap disappear, and make it seem like the monitors are of equal height.  Any suggestions?  By the way, I am also using UltraMon. Is it possible to detect 32 bit vs 64 bit in a bash script? I am writing a bash script to deal with some installations in an automated way... I have the possibility of getting one such program in 32 or 64 bit binary... is it possible to detect the machine architecture from bash so I can select the correct binary?  This will be for Ubuntu machines. "matrix is not positive definite" - even when highly correlated variables are removed How much to store luggage at SFO for 24 hours? Simple Math Problem A bat and ball cost \$1.10.  The bat costs a dollar more then the ball.  How much does the ball cost?  If this is not the correct place to ask a question like this please tell me and I will remove it :)     The ball costs \$0.05, not \$0.10 Can a shell script set environment variables of the calling shell? I'm trying to write a shell script that, when run, will set some environment variables that will stay set in the caller's shell.  setenv FOO foo   in csh/tcsh, or  export FOO=foo   in sh/bash only set it during the script's execution.  I already know that   source myscript   will run the commands of the script rather than launching a new shell, and that can result in setting the "caller's" environment.  But here's the rub:  I want this script to be callable from either bash or csh.  In other words, I want users of either shell to be able to run my script and have their shell's environment changed.  So 'source' won't work for me, since a user running csh can't source a bash script, and a user running bash can't source a csh script.  Is there any reasonable solution that doesn't involve having to write and maintain TWO versions on the script? Can you use Magical Secrets to choose high level spells when multiclassing as a Bard? Say you have a Wizard 7/Bard 9. You gain your 17th level and choose to gain the level in Bard (which grants you Magical Secrets). According to the Multiclass Spellcaster: Spell Slots per Spell Level table, you now have a single 9th level spell slot. Can the spells you choose from Magical Secrets be 9th level spells even though you can only cast up to 5th level Bard spells? Prove Borel Measurable Set A with the following property has measure 0. This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition.  Let $\epsilon \in (0,1)$, let $m$ be Lebesgue measure, and suppose $A$ is Borel Measurable subset of $\mathbb R$. Prove that if $$ m(A\cap I)\leq (1-\epsilon)m(I)$$  for every interval $I$, then $m(A)=0$.  Collection of my thoughts  1. Try to prove by contradiction, suppose $m(A)=a&gt;0$ Borel Measurable Set satisfies the condition given. A property of any Borel Measurable Set on $\mathbb R$ is $A$ contains a closed set $F$ and is contained in open set $O$, and the measure of $F$ and $O$ can be arbitrarily close to $a$. 2. Obviously $A$ contains no interval, but it does not lead to contradiction, because we can find an example of $m(A)&gt;0$ that contains no interval(fat Cantor Set(remove middle 1/4 each time) has measure $1/2$ and contains no interval).  2'. For fat Cantor Set $C$ and given $\epsilon&gt;0$ how could we construct an interval I satisfies $m(C \cap I)&gt;(1-\epsilon)m(I) ?$   Maybe try to write Borel measurable set as countable union/intersection of intervals.   I'm stucked here and will appreciate any help, some examples/special case will be great too. How can gravity affect light? Number of homomorphisms from $(\mathbb Q^+ , \,\cdot\,)$ to $(\mathbb Z_n , +)$ How many homomorphism are there from $(\mathbb Q^+ , \,\cdot\,)$ to $(\mathbb Z_n , +)$?   Could anyone give a hint how to proceed? I couldn't even start Elevating process privilege programmatically? I've bent (again) my derailleur hanger. It's a replaceable hanger but for this summer I can't get an original one and I know that the non-original one I can get is made in a harder material and could possibly damage the frame.  So I'm left with the choice of straightening this light aluminium alloy thing. How do I do that without breaking/weakening it?