Prove if $a_{n}$ converge to $L$ then $\lim_{n\rightarrow\infty}\frac{a_{1}+a_{2}+...+a_{n}}{n}=L$
Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means
There is a statement that maximizing the likelihood is equivalent to minimizing the cross-entropy.  Are there any proof for this statement? XeLaTeX font rendering tend to be slightly bolder compared to PDFlatex What's a good phrase for "refining a process which is hopelessly broken"? I'm looking for a turn of phrase to describe a situation where the powers that be wish to continue making small improvements to a process which, due to deep-rooted flaws, will never be close to accurate.  I'm working on a process for estimating the states of business objects, and there's interest in fixing some edge cases which have been noticed.  This could be worthwhile, but because of missing data we're already making heroic guesses.  So if implemented, we might get from 80.0 to 80.1% accuracy.  If we were at 99.5%, I'd be happy to get it to 99.6%, but in the current situation it doesn't seem productive.  We have some CS people on E.SE; what would you call it when someone wants to optimize a two-second process to load data, though once loaded it will be chewed on by an inefficient algorithm for hours?  Something like "bailing the ocean" or "fixing a leaky faucet in a burning building" is close to what I have in mind.  There's also "rearranging deck chairs on the Titanic," but that carries an implication that the changes are wholly useless, and that events are coming to a head. Finding a basis for a certain vector space of periodic polynomials Where do the regions of Westeros appear to align as of the season 6 finale? Based on , my understanding is that there are 9 regions of Westeros subject to the rule of the Iron Throne, from Dorne joining up to Cersei killing Robert.  With whom does [insert region] align in what appears to be Daenerys vs Cersei?   Kingdom of the North -- Daenerys probably but currently busy with White Walkers Kingdom of the Mountain and the Vale -- Daenerys probably but currently busy with White Walkers (supposedly; depends on what Littlefinger has in mind) Kingdom of the Isles and Rivers (Iron Islands) -- Daenerys Kingdom of the Isles and Rivers (Riverlands) -- Cersei because the Freys are with the Lannisters? Kingdom of the Rock -- Cersei because she is a Lannister Kingdom of the Stormlands -- Cersei? Kingdom of the Reach -- Daenerys after invitation of Varys and Ellaria Principality of Dorne -- Daenerys after invitation of Varys presumably Crownlands -- Cersei since Cersei is currently occupying the Iron Throne   Am I understanding this right? Anything wrong? Preseed Ubuntu 16.04 not working (preseed file not found) I've been trying to preseed my installation for 2 days now. I'm quite new in this topic, so execuse me if I'm asking noob questions. How do I use boot parameters, how do I know that my preseed file works. When is my preseed file loaded In US hotels, what happens if you skip checkout and just leave the hotel? Every time I check out of a hotel, they ask me if my stay went fine, let me review my incidental charges and ask if they should charge the credit card used to check in.   But what if I simply walk out of the hotel without formally "checking out"? They have my credit card on file and these days you don't even have to return your room key if you want to keep a souvenir. Would it be rude and create more work for hotel employees?  The question arose after spending 15 minutes in the check out queue at a major US hotel. Want to fill line with repeating string Similar to the \dotfill command, but with your own text, rather than dots.  If the string was: kitty!  The entire line would look like the following: kitty!kitty!kitty!kitty!kitty!kitty!kitty!kitty!kitty!  Is there any way to do this? Choosing OpenLayers or Leaflet? I was debating with one of my collegues on OpenLayers v/s Leaflet. I made a point that OpenLayers is much better API if we wish to build a project, where you need direct connectivity to the Geoserver and PostGIS.  Then I found , which looks pretty new but has the features of connectivity with the Geoserver and PostGIS.  So my project details are as follows,   Use the map interface to fetch Feature Info Create a customized tool that takes the lat/lon from user as to where he/she clicks on the map and then fetches the Climate Data from the raster (which is handled by a py script on the server) Allows user to upload excel, which is sent to the py script, which returns a GeoJSON, which creates Vector Features on the map Allow user to create vector polygons, which will fetch the Features it intersects from the WFS Layer. Fetches Layer from the PostGIS Datastore on GeoServer and displays the layers on the map.   So now I am confused on which is better and why using OpenLayers over Leaflet makes more sense or not? What's the simplest way to decrement a date in Javascript by 1 day? I need to decrement a Javascript date by 1 day, so that it rolls back across months/years correctly.  That is, if I have a date of 'Today', I want to get the date for 'Yesterday'.  It always seems to take more code than necessary when I do this, so I'm wondering if there's any simpler way.    What's the simplest way of doing this?  [Edit: Just to avoid confusion in an answer below, this is a JavaScript question, not a Java one.] How can I install WhatsApp in Ubuntu? Related: In An Introduction to Statistical Learning (James et al.), in section 3.7 exercise 5, it states that the formula for $\hat{\beta}_1$ assuming linear regression without an intercept is  $$\hat{\beta}_1 = \dfrac{\displaystyle\sum\limits_{i=1}^{n}x_iy_i}{\displaystyle \sum\limits_{i=1}^{n}x_i^2}\text{,}$$ where $\hat{\beta}_0 = \bar{y}-\hat{\beta}_1\bar{x}$ and $\hat{\beta}_1 = \dfrac{\displaystyle S_{xy}}{S_{xx}}$ are the usual estimates under OLS for simple linear regression ($S_{xy} = \displaystyle\sum\limits_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})$).  This is not the actual exercise; I am merely wondering how to derive the equation. Without using matrix algebra, how do I derive it?   My attempt: with $\hat{\beta}_0 = 0$, we have $\hat{\beta}_1 = \dfrac{\bar{y}}{\bar{x}} = \dfrac{S_{xy}}{S_{xx}}$.  After some algebra, it can be shown that $\displaystyle S_{xy} = \sum\limits_{i=1}^{n}x_i^2 - n\bar{x}\bar{y}$ and $\displaystyle S_{xx} = \sum\limits_{i=1}^{n}x_i^2 - n\bar{x}^2$. From here, I'm stuck. What registers to save in the ARM C calling convention? It's been a while since I last coded arm assembler and I'm a little rusty on the details. If I call a C function from arm,  I only have to worry about saving r0-r3 and lr, right?  If the C function uses any other registers, is it responsible for saving those on the stack and restoring them? In other words, the compiler would generate code to do this for C functions.   For example if I use r10 in an assembler function, I don't have to push its value on the stack, or to memory, and pop/restore it after a C call, do I?  This is for arm-eabi-gcc 4.3.0. Professionals: Would you choose GIMP over Photoshop and/or Inkscape over Illustrator? On convergence of Bertrand series $\sum\limits_{n=2}^{\infty} \frac{1}{n^{\alpha}\ln^{\beta}(n)}$ where $\alpha, \beta \in \mathbb{R}$ Study the convergence of $$\sum_{n=2}^{\infty}  \frac{1}{n^{\alpha}\ln^{\beta}(n)}$$ where $\alpha, \beta \in  \mathbb{R}$   I have proved that:   This series diverges when $\alpha \leq 0$. This series converges when $\alpha &gt; 1, \beta &gt; 0$ This series diverges when $0 &lt; \alpha &lt; 1, \beta &gt; 0$ This series converges when $\alpha = 1, \beta &gt; 1$      Question: What happens when $\alpha &gt; 0$ and $ \beta &lt; 0$?   There are other questions on MSE which ask about this series, but this question is distinct because   I would like an argument which does not rely on the integral test for series convergence, and this question considers all real $\alpha$ and $\beta$, while other questions ask only about $\alpha, \beta &gt; 0$, where we can apply Cauchy condensation criterion What to do after visiting unsafe site How to prove that a simple graph having 11 or more vertices or its complement is not planar? It is an exercise on a book again. If a simple graph $G$ has 11 or more vertices,then either G or its complement  $\overline { G } $ is not planar.  How to begin with this? Induction?  Thanks for your help! Interpret geometrically the equation The function $f$ is convex when:  $$f(tx+ (1-t)x', ty + (1-t)y') ≤  tf(x,y) + (1-t)f(x', y')$$  for all numbers $0≤t≤1$ and all pairs of points $(x,y)$ and $(x',y')$ in the domain of $f$ (the whole of $R^2$)  How would I interpret this geometrically? Also what is an example of a convex function whose graph is not a plane? Difference between shadowing and overriding in C#? Date Format In RUBY