How to find the limit of this sequence. I tried using the squeeze theorem but no result yet. $ \lim_{n\to \infty}\sqrt[n]{\frac{2^n + 3^n + 4^n}{5^n + 6^n}}$
I came across this strange limit whilst showing convergence of a series: $$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$$ How can I calculate this limit?
How do I diff two text files in Windows Powershell? I have two text files and want to find the differences between them using Windows Powershell. Is there something similar to the Unix diff tool available? Or is there another other way I haven't considered?  I've tried compare-object, but get this cryptic output:  PS C:\&gt; compare-object one.txt two.txt  InputObject                                                 SideIndicator -----------                                                 ------------- two.txt                                                     =&gt; one.txt                                                     &lt;= Where can I find the status of the released versions of Ubuntu? What is the width of the standard paragraph indent in LaTeX's article class? If no changes has been made to the settings of indentation of an article class LaTeX document, what is the width of the paragraph indent?  texdoc article in the terminal opens the Standard Document Classes for LaTeX version 2e manual. In section 6.2 Paragraphing I find the following line:  ⟨10pt⟩ \setlength\parindent{15\p@}   The  means 1pt ... so I guess this would mean that the standard indent is 15 pt at a 10 pt text size? Is this correct? Can a .NET windows application be compressed into a single .exe? Is the .minecraft folder hidden on a Mac? Creating intuition about Laplace & Fourier transforms Minecraft Survival Mode I am playing Minecraft on survival mode. I can break blocks like sand, dirt, and gravel and it will give me a block. When I try to break stone or iron or anything else, it didn't give me a block. What can I do so that I can get a block? I am on single player and my internet is fine. I tried deleting the app and then getting the app again but that didn't work. This is happening with all my worlds in survival mode. Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. Definition of autocorrelation time (for effective sample size) I've found two definitions in the literature for the autocorrelation time of a weakly stationary time series:  $$ \tau_a = 1+2\sum_{k=1}^\infty \rho_k \quad \text{versus} \quad \tau_b = 1+2\sum_{k=1}^\infty \left|\rho_k\right| $$  where $\rho_k = \frac{\text{Cov}[X_t,X_{t+h}]}{\text{Var}[X_t]}$ is the autocorrelation at lag $k$.    One application of the autocorrelation time is to find the "effective sample size": if you have $n$ observations of a time series, and you know its autocorrelation time $\tau$, then you can pretend that you have  $$ n_\text{eff} = \frac{n}{\tau} $$  independent samples instead of $n$ correlated ones for the purposes of finding the mean.  Estimating $\tau$ from data is non-trivial, but there are a few ways of doing it (see ).  The definition without absolute values, $\tau_a$, seems more common in the literature; but it admits the possibility of $\tau_a&lt;1$.  Using R and the "coda" package:  require(coda) ts.uncorr &lt;- arima.sim(model=list(),n=10000)         # white noise  ts.corr &lt;- arima.sim(model=list(ar=-0.5),n=10000)    # AR(1) effectiveSize(ts.uncorr)                             # Sanity check     # result should be close to 10000 effectiveSize(ts.corr)     # result is in the neighborhood of 30000... ???   The "effectiveSize" function in "coda" uses a definition of the autocorrelation time equivalent to $\tau_a$, above.  There are some other R packages out there that compute effective sample size or autocorrelation time, and all the ones I've tried give results consistent with this:  that an AR(1) process with a negative AR coefficient has more effective samples than the correlated time series.  This seems strange.    Obviously, this can never happen in the $\tau_b$ definition of autocorrelation time.  What is the correct definition of autocorrelation time?  Is there something wrong with my understanding of effective sample sizes?  The $n_\text{eff} &gt; n$ result shown above seems like it must be wrong... what's going on? Suppose I know that I can find the $\gcd$ of two integers with the Euclidean algorithm. What is the 'extension' so I can use this method also for polynomials?  Edit: I found an easy answer with less rigor which is approachable without knowledge of matrices:    also the first 3 subchapters of MySQL returns only one row How to identify whether a file is normal file or directory What is the population of the Star Wars Galaxy? I have the following code and I expect the resulting angle to be 135 degrees but whatevet I try, I get 90.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \coordinate (A) at (1,0); \coordinate (B) at (-1,1);  \pgfmathanglebetweenpoints{A}{B} \edef\angleAB{\pgfmathresult} \node at (0,0) {\angleAB};  \end{tikzpicture} \end{document}   I also tried the following command without knowing what it means, it also prints 90 except one case. The only working case is when A=(1,0) and B=(-1,0), which it prints 180.  \pgfmathanglebetweenpoints{\pgfpointanchor{A}{center}}{\pgfpointanchor{B}{center}}   Why is that? Couldn't find any reference document for "pgfmathanglebetweenpoints" by the way, that's why I am asking here.  EDIT: B is corrected from (-1,-1) to (-1,1) Open a window on a remote X display (why "Cannot open display")? Once upon a time,  DISPLAY=:0.0 totem /path/to/movie.avi   after ssh 'ing into my desktop from my laptop would cause totem to play movie.avi on my desktop.  Now it gives the error:   No protocol specified Cannot open display:    I reinstalled Debian squeeze when it went stable on both computers, and I guess I broke the config.   I've googled on this, and cannot for the life of me figure out what I'm supposed to be doing.   (VLC has an HTTP interface that works, but it isn't as convenient as ssh.)  The same problem arises when I try to run this from a cron job. I read in a book &quot;Physics - Resnik and Halliday&quot; the explanation of Type-I Superconductors {cold ones} that:  The Electrons that make up current at super-cool temperatures move in coordinated pairs. One of the electrons in a pair may electrically distort the molecular structure of the superconducting material as it moves through, creating  nearby short-lived regions of positive charge.the other electron in the the pair may be attracted to the positive spot. According to the theory the coordination would prevent them from colliding with the molecules of the material and thus would eliminate electrical resistance  Is this the only explanation or can somebody give me a more intuitive explanation that also takes into the problem of defect scattering as in the case of resistance and also explains the Type-II superconductors {hot ones} P.S. What are &quot;coordinated pairs&quot;? Do iOS app visits count to your consecutive visits Is the graviton hypothetical? Wikipedia lists the  as a hypothetical particle. I wonder whether graviton is indeed hypothetical or does its existence directly follow from modern physics? Does observation of gravitational waves amount to the discovery of graviton or could there be gravitational waves not composed of gravitons? Can I use a HDD on my PS3 if I have already used another on the same console? I'm thinking of changing the Hard Disk of my PS3 (Slim 120GB), but I am unsure of what will happen to the data on the one I previously had if I do this:   PS3 has some data in the HDD. I turn off PS3, change HDD, install OS on new HDD. I download some data on new HDD from PS Plus.   Let's imagine I forgot to back something up...   I turn PS3 off, change HDD back.   Can I still use the old hard drive without needing to format it again? How can I drive 12v LEDs from my Arduino Nano? Background I am building a scale RC helicopter and ordered a bunch of 3mm LEDs to give it some nice navigation and strobe lights which I plan to drive with an Arduino Nano. Problem is that I goofed and didn't notice that the LEDs are 12v until they arrived.  I know that the Arduino Nano can accept 12V on pin 30, but I think the voltage output on any of the &quot;D&quot; or &quot;A&quot; pins is always 5v (correct??). Question If the voltage output of the Arduino pins is 5v, then how can I drive the 12v LEDs using the Arduino, or is it impossible?