Proof of $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$
Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
How do I efficiently filter computed values within a Python list comprehension? The Python list comprehension syntax makes it easy to filter values within a comprehension.  For example:  result = [x**2 for x in mylist if type(x) is int]   Will return a list of the squares of integers in mylist.  However, what if the test involves some (costly) computation and you want to filter on the result?  One option is:  result = [expensive(x) for x in mylist if expensive(x)]   This will result in a list of non-"false" expensive(x) values, however expensive() is called twice for each x.  Is there a comprehension syntax that allows you to do this test while only calling expensive once per x? Prove F test is equal to T test squared I need to show that F test is equal to T test squared, when the T test is for 2 independent groups and assuming variances are equal.  I know that $F=\frac{MSB}{MSW}=\frac{SSB/k-1}{SSW/N-K}$ and I know that $T=\frac{X-Y}{S_p \sqrt{\frac{1}{n}+\frac{1}{m}}}$,   so $T^2=\frac{(X-Y)^2}{S_p^2 ({\frac{1}{n}+\frac{1}{m}})}$  I've seen this proof in Regression but here we're not using MSE and MSR, so i'm not sure how to connect between the two. Suppose that $f$ is analytic on a close curve γ. Prove or disprove $\int_\gamma \overline{f(z)}f'(z)dz$ is purely imagine. Suppose that $f$ is analytic on a close curve γ. Prove or disprove   $$\int_\gamma \overline{f(z)}f'(z)dz$$  is purely imagine.  I know that $f$ is analytic on a close curve, then   $$\int_\gamma f(z) dz=0$$  I tried an example with $\gamma =e^{it}$ with $0\leq t\leq 2\pi$, I often get the real part of the integral equal zero.  I tried let $f=u+iv$ so $f'=u_x+iv_x$. Since $f=u+iv$, $\overline{f}=u-iv$  $$\overline{f(z)}f'(z)=uu_x+vv_x+i(uv_x-vu_x)$$  But this doesn't get me anywhere. Any help would be greatly appreciated. Game resource timers - general implementation I'm interested in understanding how resource generation timers work on many of the mmorts type games out there. I can't wrap my head exactly around how the timers are synched with servers, especially with multiple users. If I need to provide more details, please let me know.  Thanks! What's the maximum value for an int in PHP? Ignoring the special libraries that allow you to work with very big numbers, what's the largest int value you can store in PHP? How to prove that a compact set in a Hausdorff topological space is closed? So there are several types of transistors:   BJT JFET MOSFET   Combine all that with the various flavors of each (NPN, PNP, enhancement mode, depletion mode, HEXFET, etc) and you've got a wide array of parts, many of which are capable of accomplishing the same job.  Which type is best suited for which application?  Transistors are used as amplifiers, digital logic switches, variable resistors, power supply switches, path isolation, and the list goes on.  How do I know which type is best suited for which application?  I'm sure there are cases where one is more ideally suited than another.  I admit that there is some amount of subjectivity/overlap here, but I'm certain that there is a general consensus about which category of applications each of the transistor types listed (and those I left off) is best suited for? For example, BJTs are often used for analog transistor amplifiers and MOSFETs are generally used for digital switching.  PS - If this needs to be a Wiki, that's fine if someone would like to convert it for me I've only been able to resolve this far  $$ d \mid n \Rightarrow \exists l \in \mathbb{Z}\; \textrm{such that} \; n=dl \Rightarrow x^{q^n} - x = x^{q^{dl}} - x.$$  I can't move forward. Choosing the best model from among different "best" models How do you choose a model from among different models chosen by different methods (e.g. backwards or forwards selection)?  Also what is a parsimonious model? Ubuntu 17.10 windows' menu in panel I installed the latest version of Ubuntu (17.10) , but the problem is, I can't find the menu bars of applications in the GNOME panel. For instance, the Chrome menu bar doesn't appear on the panel like in Ubuntu 17.04 and earlier versions. Unmangling the result of std::type_info::name Securing WMS against unauthorized access? In a project the customer wants the images of the WMS to be secured against unauthorized access.   What are the common solutions to realize that?   Are they compatible with the OGC-standards? Error when using TikZ-Feynman package How can I setup a Material in Cycles for Z-Transparency? $\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$ Is there a way to reorganize the fields/attributes in a data extension?  I've added some extra fields that relate to other fields in that extension.   I haven't found a way to regroup those fields. Do questions about specific StackOverflow implementation details belong on SO or Meta? has several discussion comments going back and forth about whether the question belongs on SO or Meta.  The question is asking about how a specific feature of the site is implemented from a coding perspective (so that the OP might duplicate it in their own implementation), which can make it a legitimate programming question.  According to the Meta FAQ: "If your "question is about ... Stack Overflow / The core Stack Exchange engine", it should be asked here.  Here are some similar example questions on Meta:         Where do these questions belong? Is it possible to tell when statistics were last accessed by a query using T-SQL? I'm trying to ascertain what statistics are involved in a query, I know I can check the visual query plan but I wondered whether there was a table (or tables) with this information. Regex (C#): Replace \n with \r\n How can I replace lone instances of \n with \r\n (LF alone with CRLF) using a regular expression in C#?  Sorry if it's a stupid question, I'm new to Regex.  I know to do it using plan String.Replace, like:  myStr.Replace("\n", "\r\n"); myStr.Replace("\r\r\n", "\r\n");   However, this is inelegant, and would destroy any "\r+\r\n" already in the text (although they are not likely to exist). Installing firmware blob for Intel Wifi USB pen I have installed Debian 8/Jessie minimal ISO on my laptop. I get error message during system installation:  Some of your hardware needs non-free firmware files to operate. The firmware can be loaded from removable media, such as a USB stick or floppy. The missing firmware files are: iwlwifi-7265-9.ucode iwlwifi-7265-8.ucode If you have media available now, insert it, and continue. Load missing firmware from removable media?  I found the appropriate firmware on Debian website, downloaded it (zip), extracted to flash drive, and booted with flash drive plugged in. I still get the error message that appropriate firmware could not be found (the error message disappear to fast to copy down). How do I properly format the flash drive? Or is the problem here something else? Windows Explorer on another computer tells me the flash drive is 14.6GB and formatted as FAT32.