show that the limit of $\lim_{n \to \infty} n^{\frac{1}{n}} = 1$
How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?
The total number of subsets is $2^n$ for $n$ elements In my probability book, it says that to count the total number of subsets of n elements is a process of $n$ stages with binary choice of either adding this element to the subset or not to add it. Therefore, the total number is $$2^n$$  But, for instance, we have 3 elements, according to this formula, there are 2 to the power of 3 elements, namely 8, which are $${\emptyset},A,B,C, AB, AC, BC, ABC$$  However, I have a hard time of imagining the process or N stages binary choice that form this many subsets. Can anyone explain/help me to understand it? I mean, ABC, if we are making the choice of A, put it in or do not put it in, exactly which subset are we choosing to put in or not? Thank you. What is the best way to detect if a user leaves a web page?  The onunload JavaScript event doesn't work every time (the HTTP request takes longer than the time required to terminate the browser).  Creating one will probably be blocked by current browsers. Answer appears automatically converted as a comment I once learnt, that it is uncool to use $$, but why is that? Why does \[ not act as a abbreviation to \begin{align}? I noticed, that there is a difference, since one cannot use &amp; and \\ inside a block started with \[. $5$ tourists plan to attend Octoberfest. Each attends a location at random from the choices: Alpine, Bingeman, Concordia, Kitchener, Queens, Schwaben, Transylvania. Q What is the probability that two attend one location and three attend another same location?  I get that there are $\binom{5}{2}$ possibilities to create 2 groups. There are $\binom{7}{2}$ ways to choose 2 places so shouldn't it be $\frac{\binom{7}{2}\binom{5}{2}}{7^5}$? What files should i put in a logo pack to give to clients? Should I put small, medium and large version of my logo as well as the vector file. How do I create a very long delay with Redstone? The relationship between the eigenvalues of matrices $XY$ and $YX$ If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove the following?  $$\sigma (XY) = \sigma(YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$$  Here, $\sigma$ denotes the set of eigenvalues/spectrum. Do conjugate priors just lead to a posterior that is a modification of the parameters of the prior? We know, for example, that the conjugate relationship between the classic beta-binomial is as follows:  \begin{align}  y &amp;∼ \mathcal{Bin}(n,\ θ)  \\  θ &amp;∼ \mathcal{Beta}(α,\ β)  \\ θ|y &amp;∼ \mathcal{Beta}(y + α,\  n − y + β) \end{align}  Notice how the posterior is just a change in parameterization compared to the prior. Is the posterior of conjugate priors just a change in parameters? Rooks on a 8 by 8 checker board. Probability problem Two rooks are placed on an 8 by 8 chessboard such that they're in different squares. What is the probability they can't attack each other? Rooks can attack each other if and only if they're in the same row or column. I know that there are questions like this but they give the general numbers not the probability I am not sure how to start the questions and I have looked at this questions for hours. Any help would be welcome. is it possible to redefine { and } in C? Math Induction Proof: $(1+\frac1n)^n < n$ Can I kill the Golem pre-hardmode? How to prove the divisibility rule for $3\, $ [casting out threes] The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three.  This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:  $$1212582439 \rightarrow 37 \rightarrow 10\rightarrow 1 \implies 3\not\mid 1212582439$$ $$124524 \rightarrow 18 \rightarrow 9 \implies 3\mid 124524$$  This works for as many numbers as I've tried.  However, I'm not sure how this may be proven.  Thus, my question is:     Given a positive integer $n$ and that $3\mid\text{(the sum of the digits of $n$)}$, how may we prove that $3\mid n$? Arrive steering behavior Background I'm working on a time series data set of energy meter readings. The length of the series varies by meter - for some I have several years, others only a few months, etc. Many display significant seasonality, and often multiple layers - within the day, week, or year. One of the things I've been working on is clustering of these time series. My work is academic for the moment, and while I'm doing other analysis of the data as well, I have a specific goal to carry out some clustering. I did some initial work where I calculated various features (percentage used on weekends vs. weekday, percentage used in different time blocks, etc.). I then moved on to looking at using Dynamic Time Warping (DTW) to obtain the distance between different series, and clustering based on the difference values, and I've found several papers related to this. Question Will the seasonality in a specific series changing cause my clustering to be incorrect? And if so, how do I deal with it? My concern is that the distances obtained by DTW could be misleading in the cases where the pattern in a time series has changed. This could lead to incorrect clustering. In case the above is unclear, consider these examples: Example 1 A meter has low readings from midnight until 8AM, the readings then increase sharply for the next hour and stay high from 9AM until 5PM, then decrease sharply over the next hour and then stay low from 6PM until midnight. The meter continues this pattern consistently every day for several months, but then changes to a pattern where readings simply stay at a consistent level throughout the day. Example 2 A meter shows approximately the same amount of energy being consumed each month. After several years, it changes to a pattern where energy usage is higher during the summer months before returning to the usual amount. Possible Directions  I've wondered whether I can continue to compare whole time series, but split them and consider them as a separate series if the pattern changes considerably. However, to do this I'd need to be able to detect such changes. Also, I just don't know if this is a suitable way or working with the data. I've also considered splitting the data and considering it as many separate time series. For instance, I could consider every day/meter combination as a separate series. However, I'd then need to do similarly if I wanted to consider the weekly/monthly/yearly patterns. I think this would work, but it's potentially quite onerous and I'd hate to go down this path if there's a better way that I'm missing.  Further Notes These are things that have come up in comments, or things I've thought of due to comments, which might be relevant. I'm putting them here so people don't have to read through everything to get relevant information.  I'm working in Python, but have rpy for those places where R is more suitable. I'm not necessarily looking for a Python answer though - if someone has a practical answer of what should be done I'm happy to figure out implementation details myself. I have a lot of working &quot;rough draft&quot; code - I've done some DTW runs, I've done a couple of different types of clustering, etc. I think I largely understand the direction I'm taking, and what I'm really looking for is related to how I process my data before finding distances, running clustering, etc. Given this, I suspect the answer would be the same whether the distances between series are calculated via DTW or a simpler Euclidean Distance (ED). I have found these papers especially informative on time series and DTW and they may be helpful if some background is needed to the topic area: Short circuit evaluation and side effects Whoa! What is this "Linked" thing? As per the title, what is this new feature? OpenLayers and OSGB 1936 (CRS 27700) I've been working with loads of shapefiles projected according to OSGB 1936 (CRS 27700). The project CRS is set to EPSG:27700 and "On the fly reprojection" to EPSG:27700 is set (I keep this as a default as I work with OSGB 1936 95% of my time).  I decided to download OSM layers from the OpenLayers plugin. When I load this, the EPSG in the bottom right of the screen goes from EPSG:27700 to EPSG:3857   Question 1: should this be happening anyway as it would suggest that the whole project has been re-projected to EPSG:3857?   As soon as the OSM layer is loaded it's way out of alignment but a quick pan seems to bring it altogether quiet nicely.   What I have noted is that (Question 2), if I try to re-specify the CRS to 27700, then the shapefiles "move" by a few tens of meters with respect to the OSM. Does this suggest that QGIS is doing a second re-projection (from what to what?)?   Also (Question 3), does it mean that if I import another shapefile that would be originally a dataset digitised with EPSG:27700 it would be re-projected to EPSG:3857 because the CRS in the bottom right of the screen was set to EPSG:3857 after the OSM import. This is all a bit confusing and logic would suggest that the import of a non-ESPG:27700 (project default CRS) should not go and change the EPSG setting in the bottom right of the screen. Anita Graser's book says on page 34 "Your project CRS will automatically be changed to Pseudo Mercator when you load a background map using the OpenLayers plugin option".   Does that mean there is no way of keeping ESPG:27700 active in the project when importing basemaps from OpenLayers? If so, what alternatives are there?  What I am trying to bottom out is to understand the correct way of importing basemaps (usually Google, Bing or OSM through the OpenLayers plugin) into my projects (always EPSG:27700) without losing alignment and getting layers to correctly re-project. I'm trying to support greater use of QGIS in my company but don't want to lose credibility by promoting the use of a tool that is misaligning datasets. Don't get me wrong. QGIS is a great tool and I am very fond of it but importing layers with different CRS's is quite common these days and getting this right from the start is half the job done.  I did quite an extensive Google search before posting and didn't come up with a recent, clear cut answer to my questions above. There is a bit of content out there on problems with older versions of QGIS, PROJ4, etc. but nothing recent for QGIS 2.0 so worth an update. Labels cannot be moved although data-defined settings are active I'm trying to manually move labels but having some trouble. I'm trying to move the labels while in edit mode. I've added fields for x coordinate (all null), y coordinate (all null), rotation (all =0), and font-size (all =10). This makes the "move label" and "rotate label" buttons active, but when I click on the labels, no box around the label appears and it's impossible to move the label. What am I doing wrong?