If $x_n \to a$ then $$\lim_{n \to \infty}\frac{x_1+x_2+x_3+...+x_n}{n}=a$$  My attempt  $$=&gt; \lim_{n \to \infty} x_1+x_2+x_3+...+x_n = na \\ =&gt; x_1-a +x_2-a +...+x_n-a = 0 \ where \ n \ is \ not \ finite$$  Since $x_n \to a$ Given $\epsilon &gt;0$ there exists $N$ such that   $|x_n-a| &lt; \epsilon$ whenever $n&gt;N$  Applying to what we ended up with :  $n \epsilon = 0$  let $\epsilon \to 0$
Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps....
Should players get a surprise attack by interrupting a villain's speech? Just before confrontations with the "boss" of a dungeon or adventure, I often like to ham it up a bit and let the BBEG-of-the-week gloat a bit about how brilliant their plan is, or what a thorn in the side the PCs have been, or about how the PCs will never win, etc. Usually this is pretty brief, but it sets the tone of the encounter, and generally allows the players a window to say anything they wanted to say before we start the final combat. It also gives the players a chance to ask questions in case they missed something about the plot and they'd like it cleared up in-game.  Usually this goes just fine, and the players seem to generally like it, but sometimes a player who doesn't quite fit the narrative-focused mindset of the rest of the party gets a little restless and throws the first punch. Again, usually this doesn't happen, but occasionally it does.  If a player decides to do this in a given speech, how should I respond? I've defaulted to making everyone roll initiative when a player says they attack in a situation like this. After all, I don't really want to reward the players for attacking in the middle of dialogue, since I (and most of the players) find the dialogue to be really fun, and it mostly exists for the players' benefit, anyway. Besides, when the villain finally attacks, they don't get a surprise round -- their attack just means everyone has to roll initiative.  On the other hand, I feel like the player who threw the first punch might feel a little cheated in not getting a free shot off. They might be trying to have a cool moment of emulating Han Solo, blasting the enemy when they're caught off-guard gloating. Since I make them roll initiative, though, I rob them of this moment and Greedo might get the first shot off, anyway.  So, in this situation, should the player who fires first get a surprise round, or am I handling it correctly in making everyone roll initiative?  5e is the relevant edition, here. Quoting the rules is nice, but I'll take a well-reasoned argument for one way of playing or the other over a strictly RAW answer. Cycles: How do I exclude the Environment Texture from reflections? Passing a variable to sed I am looking for a good GTD (todo/task/project management) app that has cloud sync. I'd like it to be on my iPhone, iPad and Mac. Web app would be nice, too.  OmniFocus is the closest thing I've found but it seems too expensive.  Things don't have clound sync... only WiFi sync is available for Things.  Do you know of any GTD app I'm looking for? Thanks! Remove Punctuation When Checking for Duplicate Titled Questions I recently saw a question titled "" (10k+ users only). I thought, "surely someone has asked this question before", and began looking for a duplicate.  is a question with the exact same title, just with a little different punctuation and casing. It was "Jquery vs Javascript" and "jQuery vs. Javascript?". The only difference being the ., ?, and the casing.  Since we now , I propose that when checking for questions with duplicate names, certain punctuation be striped out to prevent the duplicate from occurring in the first place.  As pointed out in the comments, if isn't already, white-space and non-printable characters may want to be removed for duplicate detection as well. How to create bootable ISO from an installed OS? Say I have two systems A and B.  I have installed Ubuntu 12.04 on system A and need same set of settings in system B, for which I am thinking to make a bootable iso from System A(creating as a backup). so that I can make Live USB from the generated iso and boot the System B which will be now the replica of System A.  Could you please tell is there any method or tool in doing this process? How to Create a Vinyl Plastic Toy Shader? How can I make one of the vinyl toy plastic material like this?    Video: How to safely remove proprietary Intel driver I recently installed the proprietary Intel drivers for my Toshiba Satellite A105 s4284 running Ubuntu 14.10. Nouveau was working just fine, I was just attempting to make it so that I was able to run CS:S (no need for explanation on that.)  Long story short, I went and downloaded the Intel Graphic Drivers for Linux* from . After downloading and installing the .deb, I ran intel-linux-graphics-installer which opened up a window that downloaded &amp; installed the open Intel drivers. I then rebooted, and it's slow. Very slow, especially on 3D effects and when I hit Alt+Tab.  Now, I don't know if the previously installed Nouveau drivers are conflicting with these new Intel drivers (if that's the issue, what should I do about fixing it?) What I do know is that my system isn't behaving right and I want it to (obviously.) If removing the Nouveau drivers isn't the solution, how can I safely remove the just-installed Intel drivers?  Does anyone have any suggestions? ravel: opposite meanings? Short expression to convey "but consider the source" Searching through ODT documents without opening them? Let's say I have 2000 .ODT files. Their names are random numbers. How do I go about searching for, let's say, "pricing list"? How do I find the document I need without opening all of them one by one and checking to see if it's the right one?  Is there some program to search through the words in the documents without opening them? I have a simple spreadsheet, two columns "Amount" (col Q) and "Account" (col O).  I want to get the sum of the "Amount" column where the corresponding "Account" forum equals "Paypal".  This is as far as I can get!  =SUM(Q:Q)*(O:O="Paypal") I am going to change my machine: from a laptop with Ubuntu to another laptop where I am going to install Ubuntu...  I am quite happy with the current system, what I am afraid of is having to rebuild everything from 0.  So what do people often do to clone an Ubuntu system? That may include the data, packages, shortcut settings, etc... I am trying to run a buffer tool as part of script that I will both use as a standalone script and import into ArcToolbox. I'm just wondering, if I use GetParameterAsText for the buffer distance, how do I set the linear units? E.g. I want it preset so they only have to type 300 and the script takes that to be meters? Here is what I have so far:  # Buffer the Thames feature class to create the Flood Risk Area in_features = "Thames" out_feature_class = "Flood_R_A" # Run the Buffer Tool arcpy.Buffer_analysis (in_features, out_feature_class, arcpy.GetParameterAsText(5),    "FULL", "ROUND", "ALL", "") Does assumption of normality of each mixture components implies that each margins is normal I just would like to understand some information about the joint normality and the margins. I read that the normal joint distribution almost always implies that the univariate margins are all normal.   My question is:  1- Is this apply for Gaussian mixture models? For example, in the Gaussian mixture model, each mixture component follows a Gaussian distribution. Hence, does it imply that each margin will be a Gaussian distribution as well? "I understand that mixture components are used to describe the dependency structures, and they are the joint distribution. is that correct?  2- I read this from one answer to a question in this site . "So yes, the assumption of joint normality is a sufficient condition for all marginal distributions to be normal, irrespective of the dependence structure. Hence, the theory of Copulas does not affect this result in any way."  However, I really do not understand this: irrespective of the dependence structure. As I understand if the joint distribution in normal then the dependence structure is normal.   Could someone help me with my questions, please? For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on. Disable Debug Log Programmatically I'm trying to disable WP_DEBUG_LOG if and only if debug.log exists. I'd like to hook into wp_footer and test if the file exists and if so disable my debug log. All I've found though is that WP_DEBUG_LOG is a constant and can only be changed from the wp-config.php file which is too early for my case. Is there a way to change it from a constant on install and keep it working as normal?   My only other thought is to create an option on install in the options table and just check it every time wp-config loaded to see if the value has changed but I was wanting to avoid that.   Granted I could check if the file exists before I define WP_DEBUG_LOG but somebody would have to view a page twice before debug log actually exists and is detected by the file_exists() (since it's run very early) which is not optimal.  Does anybody know of a workaround to programmatically turn off WP_DEBUG_LOG? Power Supply Capacity Formula Escaping both forward slash and back slash with sed "User accounts" or "users account"