Find the Cumulative Density Function
Find probability using distribution function method
Find The Last 3 digits of the number $2003^{2002^{2001}}$ Find The Last 3 digits of the number $2003^{2002^{2001}}$  BY number theory or otherwise,   Also i would like to ask is there a property observed in the numbers of the form $k^n$, where for some $k, n$ is varied then the digits of $k^n$ are periodic,  for example,  $2^n$,  its last digit is periodic with period 4, its second last digit is periodic $4\cdot 5 = 20$ its third last digit is periodic with periodic with period $20\cdot 5 =100$  I have observed this property with other numbers as well, though period might vary,for different values of $k$. Pumpkin Noodles is a thing or are a thing? Which of the following is correct?  How come pumpkin noodles is not a thing?  or  How come pumpkin noodles are not a thing?  I want to post one of the above statements to Twitter with a picture of a bowl of pumpkin noodles i.e. the stuff that comes out of a hollowed out pumpkin. Convergence of Ratio Test implies Convergence of the Root Test In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... I would like to take the full citation of a reference in my BibTeX file and write commentary in-line after it.  A \fullcitation{id} command would be perfect.  I would like to use it as:  \fullcitation{id1}   This paper talks about A with applications to B.  \fullcitation{id2}   This paper considers how C affects A.  Whenever the \fullcitation{id} command appears, I want it to be replaced with the full reference, including authors, title, journal, and date.  Is this possible? error: parsing file '/var/lib/dpkg/status' near line 55497 package 'indicator-displex' After running an upgrade from Ubuntu 11.10 (Oneiric Ocelot) to 12.04LTS (Precise Pangolin), a window popped up saying that the upgrade failed and that I should run sudo dpkg --configure -a.  Running that command gave me:  sudo dpkg --configure -a dpkg: error: parsing file '/var/lib/dpkg/status' near line 55497 package 'indicator-displex':  blank line in value of field 'Description   Trying to run apt-get install -f gives me many dependency errors along with that same dpkg error. I am having trouble getting a basic example of a matrix of notes to show up in my beamer presentation. I am using the code copied from the pgf manual. Here is a complete working example. The error I get upon compilation is:  "Package pgfbasematrix Error: Single ampersand used with wrong catcode."  \documentclass{beamer} \usepackage{tikz} \usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,fit,petri,positioning,matrix}  \begin{document}  \begin{frame}  \begin{tikzpicture} \matrix (magic) [matrix of nodes] { 8 &amp; 1 &amp; 6 \\ 3 &amp; 5 &amp; 7 \\ 4 &amp; 9 &amp; 2 \\ }; \draw[thick,red,-&gt;] (magic-1-1) |- (magic-2-3); \end{tikzpicture}  \end{frame} \end{document} I tried to install OracleJDK 8 using   sudo add-apt-repository ppa:webupd8team/java   But it's not working.   I searched the problem and found that      "The Oracle JDK License has changed for releases starting April 16, 2019." And  "Oracle Java (JDK) 8 Installer PPA (DISCONTINUED)"   So I installed the OpenJDK8 .   But why oracle cancelled the license of oracle 8 installer? Now that Unity is gone, the usual way of setting a Compose Key (via Keyboard Preferences > Layout) no longer works (because in Ubuntu 18.04, Devices > Keyboard has no Layout tab).  How do we set a Compose Key in Ubuntu 18.04 Bionic Beaver? Are commands complete sentences? seems to indirectly ask the question, but the upvoted answer says &quot;the actual sentence is...&quot;. So my question is this: Are commands considered to be grammatically sound, complete sentences in the English language? Some examples:  Go. Add a note. Do work. Why do shadows from the sun join each other when near enough? I was laying on my bed, reading a book when the sun shone through the windows on my left. I happened to look at the wall on my right and noticed this very strange effect. The shadow of my elbow, when near the pages of the book, joined up with the shadow of the book even though I wasn't physically touching it.   I have posted :The video seems to be the wrong way up, but you still get the idea of what is happening.  What is causing this? Some sort of optical illusion where the light gets bent?  Coincidentally, I have been wondering about a similar effect recently where if you focus your eye on a nearby object, say, your finger, objects behind it in the distance seem to get curved/distorted around the edge of your finger. It seems awfully related...  EDIT: I could see the bulge with my bare eyes to the same extent as in the video! The room was well light and the wall was indeed quite bright. How can I resize an LVM partition? (i.e: physical volume) I want to shrink my second (LVM) partition, in order to create a new partition in the newly freed space. I am using the Live CD to do so, because I know I can't resize/move this partition while it is in use. When I opened GParted in the Live CD, I realized that I could not resize the partition, because when I right-click it, the option "resize/move" is disabled.  I tried to unmount it, to "lazy" unmount it (umount -l /dev/sda2) but it didn't work.  A screenshot from GParted: "Next tag badge" panel score and answers counter discrepancy On the "Next tag badge" panel of  for C# tag, I see:      92/100 score   88/20 answers     But, when I open the "Select your next badge" panel, there's:     97/100 score   90/20 answers       I'm not sure which one is correct. But I believe that both displays should show the same counts.  The discrepancy is not due to a caching, I've noticed it the first time many weeks ago.  For these particular current counts: I have a single C# answer with 3 votes only in the last 5 days, last upvoted 13 hours ago (2015-10-12 19:30:05Z). So it should not cause the 5 votes and 2 answers difference.  I believe that the counters on the "Select your next badge" are correct and the counters on the "Next tag badge" are wrong, as I have:     Asked 1 non-wiki question with a total score of 6. Gave 90 non-wiki answers with a total score of 97.       Today (2015-10-16), the "Next tag badge" got bumped to 97/100. But "Select your next badge" shows 98/100 already. Cardinality of a totally ordered union Give us better search-by-score syntax Adding feature class to Map in CURRENT project using ArcPy in ArcGIS Pro? Flexible table with calculating column width fails starting from division by 10 I can't remember the gender of the main character.  I remember bits and pieces from different books, but I think they're all part of the same series. The pieces may not be in order or from multiple books.   Boy somehow ends up in a mirror universe that's darker and seems to have no people. He has to travel to a certain office(?) building and steal a book. I think this is towards the end of the book (I can't remember the events leading up to it). I think he finds the portal in a forest or something behind his school. He goes into the ocean with mermaids or something and has to save the ocean from pollution. Aliens exist and are observing Earth, because they think it's undeveloped and are waiting for them to be advanced enough. His mother ends up in hospital and he does something to save her. He goes to Ireland(?) and enters a mountain or something to find a race of people. I can't remember who showed him how, but I'm pretty sure she used magic. He goes to Mars and saves a race from war with themselves(?). This place was hidden inside an illusion. He's also grouped with a linguist that learned their language. He travels to another world where there's a prince. I can't remember, but I think it might've been part of a tournament. His sister goes to a robot planet through a portal. I think the actual planet was a robot/AI and had lots of little robots on it. Can't remember what happens here, but he goes there and saves her from something.   It was set a few years ago, when computers weren't common (his family got one early in the first book, I think).  Thanks in advance if someone finds it. Is there a way to detect if a browser window is not currently active? Number theory proof mod p For a given prime $p$ and integer $k$,  $k∈&lt;1, p-1&gt;$ $S(k)$ is defined as a sum of remainders of $k, k^2, k^3, ..., k^{p-1}$ when divided by $p$. Prove that $S(k)=\frac{1}{2}p(p-1)$ is true for an odd number of different k. I managed to prove that $S(p-1)=\frac{1}{2}p(p-1)$ but can't see how to do it further. "In a party people shake hands with one another"