Is there a simpler expression for $f(f(f(...f(x)...)))$ (total n '$f$') ?   Thank you.
If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use?  For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is important especially since I am trying to couple this with a limit toward infinity.
I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old.   After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.  I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality. Well, I'm trying to find the Thevenin equivalent of this circuit:    As you can see, the only current source is a dependent source. However, the control variable is in a wire where, if nothing is connected between A and B, is \$i_\Delta = 0\$ A, and that makes the source 0 A. However, if I connect an impedance \$Z\$ to the circuit, I will have current, and a Thevenin equivalent. I don't know how to solve this problem. I've bee searching here for a while, and I found this:   . I can see how is solved that case. However, I still don't know how to apply that method to my circuit.  Any help about finding the Thevenin equivalent is appreciated n.n How to test for minimum amount of items in inventory and execute a command if true I am using the AmazingKit plugin on my Minecraft server to make inventory kits for players. In a command block I have the following command:  /tellraw @p ["",{"text":"Iron","clickEvent":{"action":"run_command","value":"_____"}},{"text":" / "},{"text":"Gold","clickEvent":{"action":"run_command","value":"_____"}}]   Which prints the following result: Iron / Gold.  When a player clicks on Iron, what command should I write where the empty space is _____ in order to check if the player has at least (>=) 20 iron ingots in his inventory and if he/she does, remove 20 iron ingots and execute /kit Basic. If the player has less than 20 iron ingots in inventory, the command won't be executed. Instead, Insufficient amount of iron ingots will be printed out.   Same applies when a player clicks on "Gold".  I'm on 1.14 version How can I convert raster (stream segments result of a watershed analysis) to vector polylines, preferably shapefile. In ArcGIS this is easily acomplished with the Raster to Polyline tool.  I only see Polygonize (Raster to Vector) tool under the Raster -> Conversion menu. How do I escape a single quote ( ' ) in JavaScript? Can the Force be used to fly? The Additional Drivers settings interface is no longer available in Ubuntu 17.10  How can I install proprietary drivers in Ubuntu 17.10?  I need drivers for Wifi &amp; Bluetooth (BCM43142) and Intel Graphics.  The earlier Additional Drivers program showed a package named jockey but it no longer seem to exist in repositories. Showing inequality for harmonic series. I want to show that $$\log N&lt;\sum_{n=1}^{N}\frac{1}{n}&lt;1+\log N.$$ But I don't know how to show this. Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$ Deteremine the sum of the series $$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$$  So I first write down the $n^{th}$ term $a_n=\frac{\frac{n(n+1)}{2}}{n!}=\frac{n+1}{2(n-1)!}$.  So from there I can write the series as $$1+\frac{3}{2}+\frac{4}{2\times 2!}+\ldots +\frac{n+1}{2(n-1)!}+\ldots $$  I am quite sure I can do some sort of term by term integration or differentiation of some standard power series and crack this. Any leads? Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice. At the start of A New Hope, when the corvette is being attacked, C3PO says something to the effect of:     There'll be no escape for the princess this time.   But when Luke stumbles onto Leia's message in R2D2, C3PO seems to not know who the figure in the message is:     I'm afraid I'm not quite sure, sir. I think she was a passenger on our last voyage. A person of some importance, sir - I believe.   Is C3PO feigning ignorance to protect her, or does he really not recognize her? Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$ Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea. Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark How to change file extension at runtime in Java SEO and buying expiring or dropped domain Does it make any difference for SEO benefits to buy dropped and already available domains other than backorder domains that are expiring soon?  I heard somewhere that whois gets deleted after being dropped. But I didn't find any confirmation if it's true or not.  Anybody can tell me if there are benefits of one against the other, if yes what? How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$ I'm wondering if I have a sufficient proof of the following:   If $(a_n)$ is a sequence such that $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$.   My approach:  For all $\varepsilon &gt; 0$, there exists $N$ such that for all $k&gt;N$, $|a_k -A|&lt;\varepsilon$. So we can break the limit up as follows $$\lim_{n\rightarrow \infty} \frac{a_1+...+a_k}{n} + \lim_{n \rightarrow \infty}\frac{a_{k+1}+...+a_n}{n} \overset{\epsilon \rightarrow 0}{=} 0 + \lim_{n \rightarrow \infty}\frac{nA}{n}=A$$  Is this on the right track, or am I missing something about breaking up the limit in the way I have? What is the source for defaulting to plain English readings of non-game terms? When I try to move, rename, or delete a file named ␀in Finder, I get this error message:     The operation can't be completed, because an unexpected error occured (error code -50).   With mv, rm or find -delete in Terminal   rm: /Library/␀: Invalid argument  mv: rename /Library/␀ to /tmp/␀: Invalid argument  find: -delete: unlink(/Library/␀): Invalid argument   Can I move or delete this file in Mac OS X or do I have to use another operating system?  I even tried creating a file with this name in /tmp and it has not gone away since 8 reboots. What is the path of an MTP device in Windows? I am running Windows XP and have a phone mounted through USB by MTP.  In Explorer, its address is:  My Computer\SAMSUNG-SGH-I547\Phone\DCIM   I would like to execute a command line command such as  explorer "My Computer\SAMSUNG-SGH-I547\Phone\DCIM\Camera"   so as to quickly go to the Camera folder. (Right-clicking the device or anything in it does NOT provide a "Create Shortcut" dialog.)  But "My Computer" is not a valid PATH in Windows. Unlike USB flash drives, it seems the phone mounted by MTP does not get a letter drive name.  So, what is the path of an MTP device in Windows?  If the answer is there is no command line path, is there workaround? Getting all files that have been modified on a specific date Is it possible to find all php files within a certain directory that have been modified on a certain date  I'm using   find /var/www/html/dir/ -mtime -28 | grep '\.php'    to get files modified within the last 28 days, but I only need files that have been modified on the following date: 2011-02-08