In calculus, $\cos x$ is defined as $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}\cdots$. With this definition, how can we find the period of $\cos x$
With the Taylor series representation of $\sin$ or $\cos$ as a starting point (and assuming no other knowledge about those functions), how can one:  a. prove they are periodic?  b. find the value of the period?
Update: Failed to lock /var/cache/apt/archives/lock I am running the upgrade(12.10 to 13.04) through terminal and it keeps getting stuck at this:  Failed to lock /var/cache/apt/archives/lock   What do I do? I have root access. I have no idea what else to try other than a clean install.  Checking package manager Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done   Calculating the changes Calculating the changes  Do you want to start the upgrade?   3 packages are going to be removed. 155 new packages are going to be  installed. 1204 packages are going to be upgraded.   You have to download a total of 595 M. This download will take about  1 hour 15 minutes with a 1Mbit DSL connection and about 23 hours with  a 56k modem.   Fetching and installing the upgrade can take several hours. Once the  download has finished, the process cannot be canceled.    Continue [yN]  Details [d]y  Fetching  Could not download the upgrades   The upgrade has aborted. Please check your Internet connection or  installation media and try again. All files downloaded so far have  been kept.   Failed to lock /var/cache/apt/archives/lock    Restoring original system state  Aborting Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done  root@chaitanya-XPS-Ubuntu:~# We have two iPads, and both of them have Minecraft PE. My son wants to connect the two and play with me. I have absolutely no idea what I am doing here, so I do not know how to do this.   One game says "Broadcast to LAN", and the other one says "Local Server Multiplayer". I have switched both to on.   What do I need to do to be able to play joined? We are both on the same WiFi. Where can I find a list of Magic Items by rarity? I am currently looking for a list of Magic Items (non-homebrew and preferably Armor/Weapons) arranged in Common, Uncommon, Rare, Very Rare, and Legendary. There is not one in the Players Handbook or Dungeon Masters Guide that is appropriate.  Does such a list exist and if so, where can I find it? How did nobody at Hogwarts (especially Dumbledore) figure out what killed Myrtle? The information is all clearly there, since Harry and Ron, two 12-year-olds, were able to figure it out (with a bit of help from a petrified Hermione), thought at the time Aragog was much younger and smaller and wouldn't have been as easy to find, so maybe the "spiders flee before it" clue would have been lost, although to be honest it's not exactly as vital a piece of the puzzle as the others.  Dumbledore himself is obviously a very capable and intelligent Wizard, and after Myrtle died she came back as a ghost and was able to provide (admittedly limited) details of her own death.  I don't know about you, but I'd have thought at least someone would have tried to piece it together at the time, and it wouldn't have taken them long to figure out "Oh, she saw a pair of big yellow eyes after hearing a boy speak a funny language in the bathroom and dropped dead? Sounds like she might have heard somebody order a Basilisk to look her in the eyes in parseltongue", especially if the investigator in question was someone like Dumbledore.  Do we have any info on why nobody seemed to have found out anything in 50 years prior to Harry Potter (and, probably more importantly, Hermione Granger) turning up at Hogwarts? Batch convert 4 PDF pages to one PDF page Say we have the following sample and we are trying to estimate the variance of the sample mean of the population.  X = [0, -1, 2, 10, -3]   If I take an increasing number of bootstrap (e.g. 100, 1000, 10,000, 100,0000), the bootstrap distribution of the sample mean gets narrower and narrower. In other words, I can make the bootstrap distribution arbitrarily narrow.   Given the above, what's the point of using a different number of bootstrap samples here? And wouldn't this mean that I could make the standard error of the mean arbitrarily small by just taking more and more bootstrap samples? What am I missing?  Please note that I am doing a proper bootstrap sample in the sense that each bootstrap sample has the same exact size as the original sample (input array). In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). How to copy files into /usr/local/? (permission denied) I'm still a little new to Ubuntu and was wondering how I can copy files, in this case Aptana Studio, to /usr/local/ without being denied.   Is there a way to do it in terminal? Or a way to allow myself permission?  I should add that I am using 10.10 Desktop edition, not server. I'm looking for a free RAW editor/converter on Windows. Can you tell me some strong/weak points of them comparing with Capture NX and/or Adobe Photoshop Elements?       - conversion &amp; EXIF  - white balance  - registration not possible anymore   EXIF editing would be a nice bonus.  Related: MySQL: SELECT from another server I'm afraid that I already know the answer to my question, but I'll ask it anyway:  When there are two MySQL DB servers, can I access data that is stored on the other server?   In other words: Can I somehow do this:  INSERT INTO table (x, y, z)    SELECT x, y, x+y       FROM [otherserver].[database].[table]   Is the answer really as short as "No"? Preventing itemize environment to insert initial vertical space I'm using itemize and tables like the following  \begin{tabularx}{\textwidth}{&gt;{\ttfamily}llX}  \toprule  \textbf{\rmfamily Operator} &amp; \textbf{Typen} &amp; \textbf{Regel} \\  \midrule  = &amp; Alle Typen &amp;   \begin{itemize}    \item This is how we roll    \item Maybe we can talk?   \end{itemize} \\  \bottomrule \end{tabularx}   But it keeps inserting a vertical space before the itemized list. How can I prevent it from doing so? How do I add a repository from behind a proxy? I've a problem at the office. We're behind a proxy (which is set and applied at ubuntu proxy settings) and when I try to add a repository from terminal, I get:      Error reading --some url here--: urlopen error [Errno 113] No route to host   I've tried with   Launchpad-getkeys script. I get this (image)      I've tried with another "hack", without luck   Press Alt-F2 and type gksu gedit /usr/lib/python2.6/dist-packages/softwareproperties/ppa.py Find line 88, change keyserver.ubuntu.com to hkp://keyserver.ubuntu.com:80 Save, close and reboot.    Does anyone know if I could solve this problem in any way? Thanks Should one downvote answers to off-topic questions? It annoys me when people post answers to obviously off-topic questions like , especially after those questions have already received several close votes. Migrations seem to happen rather infrequently these days, so there's probably no reputation to gain for questions that are likely to be closed. And although sure, the answer may help the OP, it also encourages bad behavior (posting off-topic questions.)  Is it appropriate to downvote these answers?   Personally, I think I could make a case for both sides. Is it possible to get the data off of the hard drive of a PlayStation 3 that has crashed? I have already voided the warranty so I am not worried about that and I have access to quite a few tools.   I have a new PlayStation and a few saves that were not backed up. I am not worried about them but if it is possible I would love to get it moved over.   The old PlayStation will not turn on at all and I think I am going to have to pull the drive. My main concern is if I am able to transfer the files without running into any issues. I have heard that some of the stuff is specific to each PlayStation, is that true? There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? I tried to prove that $(p \to q) \lor (q \to p)$ is a tautology.  I used $p$ and $¬q$ as conditions. (Premises 1 and 5)  I managed to get to a solution, but I'm not sure if it's right.  Can you please check it?   Thank you!!!:) Why don't my female dwellers give birth to children? I am playing Fallout Shelter on my IPad. There were some women who have already gave birth, but two female dwellers are carrying their babies for about 18(!) hours. I know that this process takes three hours, so this is not normal. What do I do? Formatting each part/section/subsection Are there some magic lines to be placed in the preamble that would, like in CSS,  specify how each generic section or subsection in the whole document should be formatted? Say I want a random 1024-bit prime $p$.  The obviously-correct way to do this is select a random 1024-bit number and test its primality with the usual well-known tests.  But suppose instead that I do this:   select random odd 1024-bit number $n$ if $n$ is prime, return $n$ $n \leftarrow n+2$ goto 2   (This approach allows faster selection of primes via sieving.)  Since primes are not uniformly distributed on the number line, it would seem that this algorithm prefers primes that lie after long runs of composites.  Take a piece of the number line around $2^{1024}$ with x denoting a prime:     ---x-x----------------------------x------------------------x---x   Clearly our algorithm above is much more likely to find the 3rd prime above than to find the 2nd one.  Question: Is this a problem? Example of a finite non-commutative ring without a unity Give an example of a finite, non-commutative ring, which does not have a unity.    I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any help is appreciated.